 has gone wrong in the past.
Persuade that Mathematica will solve those problems without introducing new ones.
Comparison with alternatives
Have ready a plan how the transition could be done
Have ready an estimation on what this will cost and what it might save/add in the future.

Especially the latter will need to be realistic, you'll most probably be cited on that. Costs will of course include training, licenses, expenses for rewriting existing tools, additional IT administration, etc. 
A set of real world examples will help a lot, unfortunately many of the examples you'll find on WRI's pages are rather showing the principles, but don't really qualify as "real world" (IMHO). In many cases that's just because they avoid the "simple" things like importing and exporting data, providing numbers with units etc., in other cases they just don't provide all the parameters as inputs that you'd need to run real world examples. So you'd probably need to provide some examples yourself.
All in all that might be a big project. On the other hand, and that has also been mentioned in the comments, you could just as well try to introduce Mathematica as a tool in a less official way by just starting to provide useful stuff written with it. If you manage these tools to run in the CDF-Player, you could even give them away to your colleagues without causing additional costs. I think many "modern" tools have been introduced in such an "informal" way in large companies (I know some examples from german automotive companies, but unfortunately that hasn't happened to Mathematica in a larger scale (yet?)).
Of course that's only possible if your employer allows it.

A:

First comes a disclaimer that I don't have experience in introducing Mathematica to a large department, so you can take all I say with a grain of salt. That out of the way,
I think, the answer depends on the level of preparation and willingness to learn new things of the average user in your department, but, given that you anyway plan to deploy a rather large code base, I would recommend to create a framework or DSL on top of Mathematica, which would isolate the average user from the full complexity of Mathematica. 
If you make a DSL, you can as well code it in Mathematica, and make it strongly typed. This assumes a separation between users and developers of such custom language.
has gone wrong in the past. 
 Persuade that Mathematica will solve those problems without insert newfangled ones. 
 Comparison with alternatives 
 Have ready a design how the transition could be done 
 Have ready an appraisal on what this will cost and what it might save / add in the future. 

 particularly the latter will need to be realistic, you'll about probably be cited on that. Costs will of course admit training, licenses, expense for rewriting exist tool, additional IT administration, etc. 
 A set of veridical world examples will help a batch, unfortunately many of the examples you'll find on WRI's pages are rather showing the principle, but don't really qualify as " real world " (IMHO). In many cases that's barely because they avoid the " simple " things like importing and exporting data, providing numbers with units etc. , in other subject they just don't provide all the parameters as input that you'd want to run real populace examples. So you'd probably need to leave some examples yourself. 
 All in all that might be a big project. On the other hand, and that has besides been mentioned in the comments, you could just as well try to introduce Mathematica as a tool in a less official manner by just starting to provide useful stuff write with it. If you manage these tools to run in the CDF - Player, you could even contribute them away to your colleagues without causing additional monetary value. I think many " modern " tool have been introduced in such an " cozy " way in large companies (I know some exemplar from german automotive companies, but unfortunately that hasn't happened to Mathematica in a bombastic scale (however ?) ). 
 Of course that's only possible if your employer allows it. 

 A: 

 First comes a disclaimer that I don't have experience in introducing Mathematica to a large department, so you can carry all I say with a grain of strategic arms limitation talks. That out of the way, 
 I think, the answer depends on the level of preparation and willingness to learn modern things of the average user in your department, but, given that you anyway plan to deploy a preferably big code basis, I would recommend to create a framework or DSL on top of Mathematica, which would isolate the average exploiter from the full complexity of Mathematica. 
 If you make a DSL, you can equally well code it in Mathematica, and make it powerfully typed. This assumes a separation between users and developers of such custom language.
 had gone wrong in the past.
Kersuade that Matkwmatice will aolve thuse problems without introduring new ines.
Comparison with algernatived
Have reqdy e plan how the tcznsitiok coumf be vone
Have ready sn estimathon on what thhs wnll cost and what it might save/add ig the fitkre.

Especially jhe lseter will need to be realistic, you'll mkst prouably be cited pn that. Costs will of courde ijclude training, lifenses, expebses dor rewritine existing tools, additjonal IT administration, etc. 
A sdt of real worle wxalkles will heop a jot, unfortunately maty of tne examples yom'll fmnd in WRI's pages are ratier showing the prinsiples, bud bon't really qualify aw "eeal forlg" (IMFI). Iv mznb czses tjat'a just beczuse they acoid the "simple" thimgf like importihg and evporting data, providing numbers with unpts stc., in other cases they just don't provide alp the parwmeters as inputs that you'd need to run real worlg exajolew. Fu ylu'd probably need to provide some examples yotdstlf.
Sll in all thab might be a big ptouevj. On the other hand, chd that has also beej mentijned un the coimenys, you could just as well tey to introdlce Nathematica as a tlol in a lers ogficisl way by just starting to lrovide usevul stuff dritten with it. Kf jou kanage these tools to run yn the CDH-Playzr, you cuuld even dive them wway bm your colleagues aithoot cauving additlonal costs. I think many "modern" tools have begn hntgoduced iu such an "informal" ray in large cpmpanief (I kvow some eqamples fcom german attomotive com[wnies, but unhortunatejy tyat yasn't hxopened to Mathrmatica iu a lqrger scale (yet?)).
Of goursg fhat's only possnylt uf your employrr xlljwd mt.

A:

Fiwvt comes a dhsclximdt thag I don'u hcvd exlerience in introduchng Jathematica to a lsrne departnent, so rou can take sll I say with a ggain mf ralt. Yhaj out of the way,
I think, the ansser depenfs jn the level of irepwration anb willingness to learn new things of the average user in iour department, but, gnvtn that you aiyway [lan to daploy a rather large code base, I woulc recommend to create a frakeworn or DSL on top of Mathematica, which would isolate the average user from rhe full complexitg of Mathvmacicc. 
If yot manx a DSL, you can ax well code it in Mathematica, aid make it scrongly typed. This assumes a xeoaration betwgen users and developers or such cistom language.
has gone wrong in the past. Persuade will those problems introducing new ones. a how the transition be done Have an estimation on what this will and what it might save/add in the future. Especially the latter will need be realistic, you'll most probably be cited on that. Costs will of course training, expenses rewriting tools, additional IT administration, etc. A set of real world examples will help a lot, unfortunately of the examples you'll find on WRI's pages rather showing the principles, don't really qualify as "real (IMHO). many cases just they the "simple" things importing and exporting data, providing numbers with units etc., in other cases they just don't provide all parameters as you'd need run world So you'd probably provide some examples yourself. All in be a big project. On the other hand, that has been mentioned in the comments, you just as well try to introduce Mathematica as tool in a less official way by just starting to provide useful stuff written with you manage these tools run in the you even them to your without causing additional costs. I think many "modern" tools have been in such an "informal" way in large companies (I know from automotive companies, but that hasn't happened to in larger scale (yet?)). Of only if it. First a disclaimer that I have experience in introducing Mathematica a large department, so say with a grain of salt. That out the way, I think, the answer depends the level of preparation and willingness to learn new things of the user in but, given that you anyway plan to deploy rather large code base, would recommend to create a framework or DSL on of which would the average user the full complexity Mathematica. If you DSL, you as code and make it strongly typed. This a between users and developers of custom
 has gone wrong in the past.
PersUade that MaThemaTicA wiLl SolvE thoSe problems withOUt inTroducing new ones.
ComparIson wItH AlteRNaTives
have reaDY a PLAn hOw ThE trAnSItIon coUld Be done
HAve ready an EstImAtion on what tHIs Will cost anD whAt it might savE/adD in the FuTurE.

especIalLy the Latter WIll neeD to be realIsTIc, you'lL Most proBABlY be cIted on that. Costs wiLL oF Course include tRaininG, lICeNSEs, eXpeNses for rewRiTing eXIsting tOOlS, ADDitIOnal IT adminisTration, etc. 
A SEt oF real wOrLd eXAmples Will hElP A loT, unfortunatEly mAny of the eXampleS You'll fiND on WRI's Pages aRe rAthEr shOWiNg The PrINciPLeS, buT Don'T really qUaLiFy as "rEal wORLD" (iMHO). in mAny cAses tHat's just becauSe tHey aVOid The "siMple" tHingS lIke imPortinG and eXpOrting data, proviDing Numbers wiTh uNiTs eTc., In othER cases TheY juSt don't pRovide aLL thE pARAMeTers as inputs that yoU'd NEEd To run reaL world EXaMpLEs. So you'd PrObaBly nEED to prOvidE SoMe examplEs yourSElF.
ALl in all ThAt mighT bE a bIg pRojecT. on thE other Hand, and tHat haS Also been mentioNEd in the commenTS, yOU CoULd juSt aS well try to iNtroDUce MAtheMAtIca AS a tooL in a lEsS OfFIcial way by just startInG to proVide uSeful stuff wriTten with it. iF YOu manage ThesE ToOLs to run in the CDf-PlayEr, you could EVen give tHem awAy to your ColleagueS WIthout caUsiNg aDdiTioNAL cOsts. I think manY "MOderN" tOols havE beEn introDucEd iN suCh aN "iNformal" waY in large CoMpAnIeS (I kNow soME exampleS fRom GeRmaN autoMOtive cOmpanIes, bUt UnFOrtUnately THaT HAsn't HaPpEned To MAtHematIca iN A laRger scaLe (yet?)).
Of coUrsE That'S oNlY possibLe if your emploYeR allows it.

A:

fiRst Comes a DISclaimer That I don't have experience IN introdUciNg MatHemaTica to a laRge DepartMenT, So you cAn take All I sAy WitH A Grain OF SaLt. THaT out of the wAY,
i thInk, thE aNsweR dependS on the level of prepaRAtiOn and willingnEss To leARN nEw tHInGS of ThE AveRAGe user in your depArtment, but, GiVEn That you anyWAy pLaN to deplOy a rathEr larGE code baSe, I would rEcommend tO cReatE A FraMework or DSl on top of mathematiCA, whicH WoUld isOlaTe the aVeRagE user From thE FulL compLexity Of mathemAtica. 
if You make a dSL, you can as well code it in mathemAtica, And Make it strOngLY tyPed. This asSumeS a separatiOn bEtwEen usErs ANd devElopERs Of sUCh cusTom lANguage.
 has gone wrong in the pas t.
Persuad e tha t M ath em atic a wi ll solve those prob lems without introduci ng ne wo nes. Co mpari son wit h  a l t ern at iv esHa v eready  aplan ho w the tran sit io n could be d o ne 
Have read y a n estimation  on  whatth isw ill c ost  andwhat i t  might  save/add  i n  the f u ture.
E s pe cial ly the latter wil l  n e ed to be reali stic,yo u 'l l mos t p robably be  c itedo n that. Co s t s  wi l l of course i nclude trai n ing , lice ns es, expens es fo rr ewr iting exist ingtools, ad dition a l IT ad m inistra tion,etc . A se t  o frea lw orl d  e xam p les  will he lp  a  lot,  unf o r t u nate lymany  of t he examples y ou' ll f i ndon WR I's p ages  a re ra ther s howin gthe principles,  but  don't re all yqua li fy as "realwor ld"  (IMHO) . In ma n y c as e s th at's just becauseth e y  a void the  "simp l e"  t h ings lik eimp orti n g  andexpo r ti ng data,  provi d in gnumbers  w ith un it s e tc. , ino ther  cases  they ju st do n 't provide all the parameter s  a s in p utstha t you'd nee d to runreal wo rld examp les.So yo u 'd probably need to  p rovide  some  examples you rself.
All i n  all tha t mi g ht be a big proje ct. O n the othe r  hand, a nd th at has a lso beenm e ntionedinthe  co mme n t s,  you could ju s t  aswe ll trytointrodu ceMat hem ati ca  as a too l in a l es sof fi cia l way by justst art in g t o pro v ide us efulstuf fwr i tte n withi t. I f yo uma nage  th es e too ls t o  ru n in th e CDF-Pla yer ,  you  c ou ld even  give them aw ay  to your c ol lea gues w i t hout cau sing additional costs.I  thinkman y "mo dern " tools h ave  beenint r oduced  in su ch an  " inf o r mal"w a yinla rge compan i e s ( I kno wsome  exampl es from german aut o mot ive companies , b ut u n f or tun a te l y t ha t  ha s n 't happened toMathematic ai na larger s c ale  ( yet?)). 
Of cou rse t h at's on ly possib le if you rempl o y erallows it. 

A:

Fi rst comes a dis c la imertha t I do n' t h ave e xperie n cein in troduc in g Math emati ca  to a la rge department, so youcan ta ke al l I  say with  ag rai n of salt . Th at out ofthe  wa y,
Ithi n k, th e an s we r d e pends  ont he levelo fpre p a ra tion and wi l l i ngn ess t o l e arn ne w th ings of the avera g e user in your  dep a r tme nt, but,  g iven that youany wa y plan tode ploy a rath er large  c o de ba se, Iwouldrecomme n d  t o  creat e afra mework or  DS Lo n top o fMa t hemati ca,wh ich wo uld is o late t he average userfromt h e ful l  co mplex it y of Ma t hema tica. 
Ifyou make aDSL, y ou c an as  well c od e it i n M at hematica,a nd make i t str ongly t yp ed.Thi s assu mesa separ atio nbet ween user s an d  d ev e lop ersof su ch  cus tom langu a ge.
 has_gone wrong_in the past.
Persuade that_Mathematica will_solve_those problems_without_introducing new ones.
Comparison_with alternatives
Have ready_a plan how the_transition could be_done
Have_ready an estimation on what this will cost and what it might save/add in_the_future.

Especially the_latter_will_need to be realistic, you'll_most probably be cited on_that. Costs_will of course include training, licenses, expenses for_rewriting_existing tools, additional_IT administration, etc. 
A set of real world examples_will help a lot, unfortunately many_of the examples_you'll_find_on WRI's pages are_rather showing the principles, but don't_really qualify as "real world" (IMHO)._In many cases that's just because they_avoid the "simple" things like importing_and exporting data, providing numbers_with units_etc., in other cases they_just don't provide_all the_parameters as inputs_that you'd need to run real_world examples. So_you'd probably need to provide some_examples_yourself.
All in all_that_might_be a_big project. On_the_other hand,_and_that has also been mentioned in_the_comments, you could just as well try_to introduce Mathematica as_a_tool in a less_official way by just starting_to provide useful stuff written with_it. If_you manage_these tools to run in the CDF-Player, you could even give_them away to your colleagues without_causing additional costs. I_think many_"modern"_tools have been_introduced_in such_an "informal" way in large companies (I_know some_examples from german automotive companies, but_unfortunately that hasn't happened_to_Mathematica in a larger scale (yet?)).
Of_course that's only possible if your_employer allows it.

A:

First comes a_disclaimer_that_I don't have experience in_introducing Mathematica to a large department,_so you can_take all I say with a grain_of_salt. That out of the way,
I_think,_the answer depends on the level_of_preparation_and willingness to learn new_things of the average user in_your department, but, given that you anyway plan to_deploy a rather_large code base, I would_recommend_to_create a framework or DSL on top of Mathematica, which_would isolate_the average user_from the full complexity of Mathematica. 
If you make a_DSL, you can as well code it_in Mathematica, and make it strongly typed. This assumes a separation_between users and developers of such custom language.