Brim
Middle English (denoting the edge of the sea or other body of water): perhaps related to German Bräme ‘trimming’.
wiktionary
From Middle English brim, from Old English brim(“surf, flood, wave, sea, ocean, water, sea-edge, shore”), from Proto-Germanic *brimą(“turbulence, surge; surf, sea”), from Proto-Germanic *bremaną(“to roar”), from Proto-Indo-European *bʰrem-, *bʰerem-, *bʰrem(e)-, *breme-(“to hum, make a noise”). Cognate with Icelandic brim(“sea, surf”), Old English brymm, brym(“sea, waves”), Old English bremman(“to rage, roar”), Dutch brommen(“to hum, buzz”), German brummen(“to hum, drone”), Latin fremō(“roar, growl”, verb), Ancient Greek βρέμω(brémō, “roar, roar like the ocean”, verb).
From Middle English brim, brem, brimme(“margin, edge of a river, lake, or sea”), probably from Middle English brim(“sea, ocean, surf, shore”). See above. Cognate with Dutch berm(“bank, riverbank”), Bavarian Bräm(“border, stripe”), German Bräme, Brame(“border, edge”), 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).
See breme.
etymonline
brim (n.)
"brink, edge, margin," c. 1200, brymme "edge (of the sea), bank (of a river)," of obscure origin, perhaps akin to Old Norse barmr "rim, brim," probably related to dialectal German bräme "margin, border, fringe," from PIE *bhrem- "point, spike, edge." Extended by 1520s to the upper or projecting edge of anything hollow (cups, basins, hats).
Old English (and northern Middle English) had brim "sea, surf, pool, spring, river, body of water," of uncertain origin but probably unrelated, perhaps from the Germanic stem *brem- "to roar, rage." "It became obs. in ME.; but was perhaps used by Spenser" [OED].
brim (v.)
"to fill to the brim," 1610s, from brim (n.). Intransitive sense ("be full to the brim") attested from 1818. To brim over "overflow" is from 1825. Related: Brimmed; brimming.
