Factor causing Methicilin-resistance in MRSA?
I think you can normally think about similar pressures leading to extended-spectrum beta-lactamase (ESBLs) and  methicillin/oxacillin-resistant Staphylococcus aureus (MRSA).  

Going to the core of your question: 

"Currently, the reported mechanism of methicillin resistance in S. aureus is the production of a distinctive penicillin binding protein 2a (PBP2a), which exhibits low affinity toward β-lactams"

But to really read on it you need to back to Liu et al.

Again Todar's is a great source for a good broad reading on staph including MRSA.

Response to Edit [inclusion of full question]:

TSDR: The answer is C.

Let's break down the three options given in the question and learn about MRSA along the way.


  a. its MIC is increased to methicillin, but not to penicillin


First, for people who may not know, MIC is short for minimum inhibitory concentration.  Simply, the MIC is a basic measurement of how much of any given agent is need to stop the growth of a bacterial colony.  Often we think of current medical antibiotics when thinking of MIC, but even simpler things like table salt and sugar have a MIC for a given species.  Thus this sentence is making the assertion that the bacteria found in the patent have an increased resistance to methicillin but not to penicillin.  This leads us to ask the obvious question:

Can Staphylococcus aureus (staph going forward) be methicillin resistant but not penicillin resistant?

This is a bit of a trick question, but let's break it down to the needed components: What is Methicillin, what is/are penicillin(s), and how are staph resistant to them.

First that it should be noted that penicillins can actually refer to a whole class of antibiotics which all use β-Lactam and the specific set of antibiotics:  


  benzylpenicillin (penicillin G), procaine benzylpenicillin (procaine
  penicillin), benzathine benzylpenicillin (benzathine penicillin), and
  phenoxymethylpenicillin (penicillin V). (from wiki)


It should be noted that more correct way to address the whole class of antibiotics would be β-Lactam antibiotics, not penicillins, and that if you wanted to talk about more specifically penicillin derived compounds you would be discussing penams.  For even further clarification, penicillin as drug most likely refers to benzylpenicillin, and for the rest of this answer I will use penicillin to refer to benzylpenicillin.

Thus we should think of penicillin as an early antibodic that work by preventing the dividing/genesis of cell walls and certain organelles via binding to penicillin binding proteins (PBPs).  

Methicillin is also a β-Lactam antibiotic, and it's MOA is similar to penicillin.  It was developed/discovered after penicillin and was seen a answer to Gram-positive bacteria that were breaking down penicillin via β-lactamase.  Methicillian still works by binding to PBPs, but it escapes the bacteria's counter to penicillin.

This leads us to address how penicillin resistance and methicillin resistance commonly occur in staph.  First, many staph strains and other bacteria use β-lactamases to breakdown antibiotics so they no longer can bind to PBP's (1).  But methicillin is particularly suited by its side chains to not be degraded by β-lactamases.  In their ground breaking work on the subject, Hartman and Tomasz identified that methicillin resistance was not in the acquisition of a β-lactamase, but in a mutation in PBP's that prevented methicillin binding (2).  There they tested 4 strains of staph, two were methicillin resistant (MR), and two were methicillian susceptible (MS).  You will note that all but 1 didn't have β-lactamase activity, and where more susceptible to penicillin than methicillin (ibid).

BUT this does not mean that MR strains are not also penicillin resistant, instead it shows that the resistance can be independent of each other.  Therefore "A" is wrong because it tries to draw a correlation that is not there.  In reality many MR strains are also β-lactamase positive.

This also address the problem with "B."  While it is possible that a β-lactamase could bind to methicilin and lead to degradation of the antibiotic, the main mechanism of methicillin resistance is the mutation of PBPs (ibid, 3), in particular BPB2a (4, 5).

We are then left with the task of figuring out why "C" is the correct choice.  Indeed "C" is the given reason for the failure of "A" and "B," but we can still go deeper into how and perhaps why BPB2a mutated.

For that I think it is best that we turn back to Chambers' review of the subject.  Forgive my over quoting, but it's done so well there and the text is now open.


  Methicillin resistance is associated with production of a novel PBP
  that is not present in susceptible staphylococci. Resistant strains of
  S. aureus produce an additional 78- kilodalton PBP (Fig. 1), termed
  PBP2a or PBP2' (assumed to be identical for the purposes of this
  review), that has a low binding affinity for beta-lactam antibiotics.


...


  PBP2a is highly conserved. Limited proteolysis of PBP2a from unrelated
  strains of S. aureus (123) and coagulasenegative staphylococci (31),
  whether homogeneous or heterogeneous, generates remarkably similar
  peptide fragments.


...


  Presumably PBP2a can substitute for essential PBPs when these have
  been saturated by drug and can perform the functions necessary for
  cell wall assembly (22, 122).


...


  In some strains, PBP2a is inducible by beta-lactam antibiotics and its
  production differs according to growth conditions (34, 122, 125, 159).


Unfortunately, they didn't quite have the staphylococcal cassette chromosome mec (SCCmec) figured out at that point.  The genetics is quite complex. 

How does MRSA genetically accomplish resistance?

As we already established, the resistance comes from the production of an alternate PBP, PBP2a.  SCCmec is interesting for several reasons.  First of all, it's much larger than a plasmid, and contains information for several genes.  That's why it's called cassette chromosome.  Further it normally incorporates into the same part of the genome in staph, in an area know as OrfX (6).  This means that even during horizontal transfer, that the cassette has to direct it's integration into the genome, which is exceedingly uncommon, or at least there are not many other know examples (7, 8).  This cassette can be spread horizontally between staph, and even with other species (ibid, 9).  Even if the integration site (integration site sequence, ISS) is slightly different, this specificity is carried out by cassette chromosome recombinases (ccr), wich are also on SCCmec (8).  This is carried out by ccr-medated recombination of the target chromosome, and further details on the process are considered outside the scope of this question.

The actual gene that encodes PBP2a is called mecA.  But as we mentioned above, PBO2a production can be induced and regulated.  It is likely less favorable to produce it in the absence of antibodies, and after serial passage of bacteria in antibiotic free broth, you find that PBP2a expression can drop drastically.  Therefore regulatory and other useful proteins encoded by SCCmec.  When placed in a β-Lactamase environment, MecR1 causes a single transduction cascade to start transcription of mecA (10).  Conversely, MecI provides a negative feedback loop to MecR1, and in the absence of β-Lactamase, will lead to the down regulation of mecA (ibid, 11).  The actual action of MecR1 is to cleave MecI, thereby remove the suppression of mecA by MecI.  I actually learned something new when reading the wiki on MRSA, but didn't do further research on the subject:


  mecA is further controlled by two co-repressors, BlaI and BlaR1. blaI
  and blaR1 are homologous to mecI and mecR1, respectively, and normally
  function as regulators of blaZ, which is responsible for penicillin
  resistance. The DNA sequences bound by MecI and BlaI are
  identical; therefore, BlaI can also bind the mecA operator to
  repress transcription of mecA.


This represents the general pattern of resistance of MRSA, but there is a rich diversity in the particulars of how each strain manages expression.  Two of the main identifiers are how the mecA gene complex and ccr gene complex are configured (carriage).  The other two identifiers are the ISS and whether or not the ISS is repeated in the target chromosome (and how many times it's repeated) (8).  If we just consider the mecA and ccr carriage, then we get a great summary from IWG-SCC (ref 8):


  The mec gene complex is composed of mecA, its regulatory genes, and
  associated insertion sequences. The class A mec gene complex (class A
  mec) is the prototype complex, which contains mecA, the complete mecR1
  and mecI regulatory genes upstream of mecA, and the hypervariable
  region (HVR) and insertion sequence IS431 downstream of mecA. The
  class B mec gene complex is composed of mecA, a truncated mecR1
  resulting from the insertion of IS1272 upstream of mecA, and HVR and
  IS431 downstream of mecA. The class C mec gene complex contains mecA
  and truncated mecR1 by the insertion of IS431 upstream of mecA and HVR
  and IS431 downstream of mecA. There are two distinct class C mec gene
  complexes; in the class C1 mec gene complex, the IS431 upstream of
  mecA has the same orientation as the IS431 downstream of mecA (next to
  HVR), while in the class C2 mec gene complex, the orientation of IS431
  upstream of mecA is reversed. C1 and C2 are regarded as different mec
  gene complexes since they have likely evolved independently. The class
  D mec gene complex is composed of mecA and ΔmecR1 but does not carry
  an insertion sequence downstream of ΔmecR1 (as determined by PCR
  analysis).


And


  ccr gene complex.The ccr gene complex is composed of the ccr gene(s)
  and surrounding open reading frames (ORFs), several of which have
  unknown functions. Currently, three phylogenetically distinct ccr
  genes, ccrA, ccrB, and ccrC, have been identified in S. aureus with
  DNA sequence similarities below 50% (Fig. 2 and 3). The ccrA and ccrB
  genes that have been identified to date have been classified into four
  allotypes. In general, ccr genes with nucleotide identities more than
  85% are assigned to the same allotype, whereas ccr genes that belong
  to different allotypes show nucleotide identities between 60% and 82%.
  All ccrC variants identified to date have shown ≥87% similarity; thus,
  there is only one ccrC allotype. We suggest describing their
  differences as alleles by using previously used numbers, e.g., ccrC1
  allele 2 or ccrC1 allele 8.


All of this is nicely summarized in their table:

\begin{array} {|r|r|r|}
\hline
SCCmec ~type  &amp;ccr ~gene ~complex  &amp;mec ~gene  ~complex \\
\hline
I &amp;1 ~(A1B1) &amp;B \\
\hline
II &amp;2 ~(A2B2)&amp;A \\
\hline
III &amp;2 ~(A3B3)&amp;A \\
\hline
IV &amp;2 ~(A2B2)&amp;B \\
\hline
V &amp;5 ~(C)&amp;C2 \\
\hline
VI &amp;4 ~(A4B4)&amp;B \\
\hline
VII &amp;5 ~(C)&amp;C1 \\
\hline
VIII &amp;4 ~(A4B4)&amp;A \\
\hline
\end{array}

To sum up, methicillin-resistance is caused by the encoding of a novel PBP, PBP2a, along with other factors, which is both genetically stable and transferable via the use of SCCmec.

References:

(1) Pathak A et al. High prevalence of extended-spectrum β-lactamase-producing pathogens: results of a surveillance study in two hospitals in Ujjain, India. Infect Drug Resist. 2012;5:65-73. doi: 10.2147/IDR.S30043. Epub 2012 Apr 5. [Note that many of the great sentinel studies on drug resistant bacteria occur in the massive hospitals in India].

(2) B Hartman and A Tomasz. Altered penicillin-binding proteins in methicillin-resistant strains of Staphylococcus aureus. Antimicrob Agents Chemother. 1981 May; 19(5): 726–735. 

(3) Liu, H. et al. Detection of borderline oxacillin-resistant Staphylococcus aureus and differentiation from methicillin-resistant strains. Eur J Clin Microbiol Infect Dis. 1990 Oct;9(10):717-24.

(4) Tawil N. et al. The differential detection of methicillin-resistant, methicillin-susceptible and borderline oxacillin-resistant Staphylococcus aureus by surface plasmon resonance.  Biosens Bioelectron. 2013 Nov 15;49:334-40. doi: 10.1016/j.bios.2013.05.031. Epub 2013 Jun 4.

(5) Chambers HF. Methicillin-resistant staphylococci. Clin Microbiol Rev. 1988 Apr;1(2):173-86.

(6) Huletsky A et al.  New real-time PCR assay for rapid detection of methicillin-resistant Staphylococcus aureus directly from specimens containing a mixture of staphylococci. J Clin Microbiol. 2004;42(5):1875-84. DOI: 10.1128/JCM.42.5.1875-1884.2004

(7) Schoenfelder SM et al. Success through diversity - how Staphylococcus epidermidis establishes as a nosocomial pathogen. Int J Med Microbiol. 2010 Aug;300(6):380-6. doi: 10.1016/j.ijmm.2010.04.011. Epub 2010 May 6.

(8) IWG-SCC. Classification of staphylococcal cassette chromosome mec (SCCmec): guidelines for reporting novel SCCmec elements.  Antimicrob Agents Chemother. 2009 Dec;53(12):4961-7. doi: 10.1128/AAC.00579-09. Epub 2009 Aug 31.

(9) Stürenburg E. Rapid detection of methicillin-resistant Staphylococcus aureus directly from clinical samples: methods, effectiveness and cost considerations.  Ger Med Sci. 2009 Jul 6;7:Doc06. doi: 10.3205/000065.

(10) Jensen SO and Lyon BR. "Genetics of antimicrobial resistance in Staphylococcus aureus". Future Microbiol 4 (5): 565–82. doi:10.2217/fmb.09.30. PMID 19492967

(11) Oliveira DC, de Lencastre H.  Methicillin-resistance in Staphylococcus aureus is not affected by the overexpression in trans of the mecA gene repressor: a surprising observation. PLoS One. 2011;6(8):e23287. doi: 10.1371/journal.pone.0023287. Epub 2011 Aug 2.