What is the IUPAC name of borax?
The name suggested here (via the question comments), the anion-corresponding part, ‘bicyclo[3.3.1]tetraboroxane-3,7-diolate’ ($\ce{[B4O5(O)2]^2-}$) obviously does not match with the tetraborate formula $\ce{[B4O5(OH)4]^2-}$ present in borax, whose structure, confirmed by X-ray crystallography, including hydrogens[1], is

However, the ‘bicyclo[3.3.1]tetraboroxane’ part is correct, for the parent hydride:

This type of names is known e.g. for the ‘cyclotriboroxane’ name for boroxine, or ‘cyclotetrasiloxane’ (in octamethylcyclotetrasiloxane).
So the name would be rather 1,3,5,7-tetrahydroxybicyclo[3.3.1]tetraboroxane, or bicyclo[3.3.1]tetraboroxane-1,3,5,7-tetrol, however it still does not explicitly tell that it's an anion.
Interestingly, this boroxane, siloxane type names (interchanging element/oxygen pattern) is rather marginally mentioned in the inorganic nomenclature, notably more detailed in the organic nomenclature.[2]
Inorganic additive nomenclature is able to constructs names for similar skeletons,[3] but somewhat differently than organic nomenclature.
E.g. bicyclo[x.y.z]Nane (where N is the numerical prefix root, with the exception for 1..4, ‘meth’..‘but’, for the organic names) would be named: Ncarby-[0a.z1,b]dicycle, where a = x+y+2 and b = x+2. If the structure is an ion, it's ‘…cyclate’ followed by the charge in parentheses. One advantage is that the first numeral is the size of the (largest) cycle, not just a branch.
Some examples from the book:


1,7-diazyundecasulfy-[012.11,7]dicycle
(IR-1.5.3.4 Example 6)

cyclo-triphosporic acid
2,4,6-trihydroxido-2,4,6-trioxido-1,3,5-trioxy-2,4,6-triphosphy-[6]cycle
(IR-8.3 Example 5)

tetraaluminy-1-lithy-[05.01,301,402,5]tetracyclate(1−)
(IR-7.4.3 Example 11)

(The leading zero in the first numeral in the square brackets is intentional for the cycles, however in the latter example, it's not used, despite IR-7.4.2 Nodal descriptor, example 3, that states that the nodal descriptor for hexacycle is ‘06’.)
With this nomenclature we can get very descriptive (but almost unusable) names
1,3,5,7-tetrahydroxido-1,3,5,7-tetrabory-2,4,6,8,9-pentaoxy-[08.11,5]dicyclate(2−),
1,3,5,7-tetrahydroxidotetrabory-2,4,6,8,9-pentaoxy-[08.11,5]dicyclate(2−), or
1,3,5,7-tetrahydroxido-1,3,5,7-tetraborypentaoxy-[08.11,5]dicyclate(2−).
References:

Levy H.A., Lisensky G.C. Crystal structures of sodium sulfate decahydrate (Glauber's salt) and sodium tetraborate decahydrate (borax). Redetermination by neutron diffraction Acta Crystallographica Section B 34 (1978): p. 3507
IUPAC Blue Book 2013, P-52.1.6.2
IUPAC Red Book 2005, IR-7.4 Additive nomenclature: Inorganic chains and rings