Evidence that matrix multiplication can be done in quadratic time?
I don't know about others but I can tell you why I tend to believe that $\omega=2$. When you read through the history and development of fast matrix multiplication algorithms, I think that it is hard not to get feeling that all that we need is just a slightly better basic algorithm from which, even using known techniques, $\omega=2$ will follow. Basically, all algorithms today (including group theoretic ones) start with some "simple" construction that is then amplified. Those constructions are clever but they give the reader some sort of an "ad-hoc" feeling. I don't see a reason to believe that we cannot come up with better simple constructions.