The 126 GeV Higgs mass seems to be a rather special value, from all the a priori possible values.
One thing I've heard is that it is just at the edge of the mass range implying a stable vacuum, and that it allows vacuum stability all the way up to the Planck scale. This fact seems to be related to asymptotic safe gravity and the prediction of mH = 126 GeV by Shaposhnikov and Wetterich.
The other thing that makes me think 126 GeV is special is Supersymmetry. I was playing around with spectrum generators and exploring "simple" models like mSUGRA, and models with a bit of non-universality, and found that it is tricky to get a h0 mass as high as 126 GeV. There seems to be a glass wall or asymptote there, in that you can get very close (e.g. by choosing extreme values of A0), but often you can't arrange mh0 = 126 GeV without going to more complex models. I found a post by Lubos Motl from 2011 where he mentioned that and hinted to some interesting "marginal" properties.
So what are all the things going on at 126 GeV? What coincidences, boundaries, magical effects are there? And what makes it so hard to reach with "simple" SUSY?