Full list here, but here are a few examples: 

THIRD EYE BLIND – Our third eye is the imagined one that gives us a kind of sixth sense (telepathy, ESP, etc.) and the band felt that most of us are blind in that sense.

WEEZER – Band member Rivers Cuomo had the nickname Weezer in school because of a breathing problem.

BEASTIE BOYS – According to Michael Diamond, BEASTIE stands for Boys Entering Anarchistic Stages Towards Internal Excellence.

HOOTIE AND THE BLOWFISH – from the nicknames of two friends of singer/guitarist Darius Rucker – one with owl-like eyes (Hootie), another with the puffy “Blowfish” cheeks.