REM stands for rapid eye movement and I think it's what your eyes do whilst you are dreaming
The really interesting bit would be if someone could explain why this was a clever or interesting name for REM to have specifically. If they had hit upon a different name ("Take That", "Michael Stipe And His Band", "Doctor Livesey's Snuff-Box" or something) would that have been less clever and interesting of them?
It's tricky to see that any of these are more than just a name, which happens to sound pleasing and requires an explanation. If that's right then I'm rather with Peter that the background to the name doesn't greatly matter, except as an interesting piece of fan trivia. 🙂