"The oracle of Balaam son of Beor,
the oracle of one whose eye sees clearly . . ."
From Numbers, chapter 24. I have no idea if that's significant at all.
Apparently OE 'beor', meaning 'beer', derives from Latin 'biber', 'to
drink'. So not much in the way of significant philological depths
there. Personally I get echoes of 'Deor', ('dear'), so I suppose it's
possible Tolkien meant me to. But I think, all in all, that 'Beer' is
just a good manly name for a fighter, sounding as it does a bit like 'bear' and
'bright' and 'warrior'
also.

painting by Goldilocks Took