The Vitali set cannot be made measurable by extending the measure to *R (the Hyper-reals). One reason is that countable additivity is not (in general) defined on *R due its being Dedekind Incomplete (as noted in the first answer above). But that is not the only reason. Even if you could find a way around that (such as finding another extension of R which would allow countable additivity), the Vitali set could not be made measurable. The reason has to do with translation invariance.
As you know, the standard construction of Vitali sets yield a countable collection of sets which are congruent (modulo translation), but whose union is [0,1). So the goal here would be to assign each Vitali set a measure of some value e, such that e + e + ... = 1. However, these same Vitali sets may be rearranged (via translation only) so that its union is now [0,2). Or [0,3). Or [0,n). So now the same equation e + e + e + ... = 2, or = n, or whatever value you'd like. Translation invariance is what really kills this, not just countable additivity.
If you drop translation invariance in the measure, then you could indeed consistently assign a measure to each of the Vitali sets, so that their sum is correct. You wouldn't even need to go to *R, a translation variant measure with the standard reals could suffice to make all subsets of R measurable. Unfortunately, such measures are unsatisfying, because you are not guaranteed that the interval [x,y] always has measure y-x.
Likewise, if you drop countable additivity requirement (requiring only finite additivity), you can consistently define the Vitali set as having measure 0, and in fact assign measures to any subset of R.
The closest I have seen to reaching this goal is the Loeb Measure, a Nonstandard Measure with *R values, which has hyper-finite additivity. Still, its weaknesses and restrictions make it very unappealing (at least to me).