The Beatles was actually quite good, incorporating Beat into an existing word. (Except in German, where it apparently sounded like a rude word).

A name with some second meaning is better than one without, in my opinion.

Both solo artists and bands have gone down this road, though some references are more obscure than others. Look at Chubby Checker, for one example, and compare with Fats Domino!

I learned recently that 'Blow Monkeys' is slang for saxophone players.
(I have led a sheltered life).

Have a brain-storming session where all suggestions, however silly, are considered. They can often lead on to something unexpectedly good.