Alpha is actually made up of several constants, as shown in the wikipedia article. So, the question is, if this is indeed the case that alpha isn't constant, which of these 'constants' is actually not a constant?
e is the elementary charge. The charge on a proton (-e for an electron). Somehow I think this is unlikely not to be a constant as for all intents and purposes all protons are the same as any other proton, same with electrons.
h is the Planck constant, which relates energy to frequency of electromagnetic waves, for example. I'd say that it's a relational constant to create different ways of saying the same thing, so I wouldn't think this is a variable.
c is the speed of light in vacuum, 0 is the permittivity of free space, 0 is the magnetic constant or permeability of free space. All three are related by Maxwell's laws. My guess is that it might be one (or all, or some) of these that would be the most likely to not be a variable.
Of course, as with the faster-than-light neutrinos, we'll just have to wait for the results to be checked before we can jump to any radical conclusions...