From Middle English brim, from Old English brim, from Proto-Germanic *brimą, from Proto-Germanic *bremaną, from Proto-Indo-European *bʰrem-, *bʰerem-, *bʰrem(e)-, *breme-. Cognate with Icelandic brim, Old English brymm, brym, Old English bremman, Dutch brommen, German brummen, Latin fremō, Ancient Greek βρέμω.
From Middle English brim, brem, brimme, probably from Middle English brim ("sea, ocean, surf, shore"). See above. Cognate with Dutch berm ("bank, riverbank"), Bavarian Bräm ("border, stripe"), German Bräme, Brame, Danish bræmme ("border, edge, brim"), Swedish bräm ("border, edge"), Icelandic barmur ("edge, verge, brink"). Related to berm.
Either from breme, or directly from Old English bremman ("to roar, rage") (though not attested in Middle English).