brim n.

"brink, edge, margin," c. 1200, brymme "edge (of the sea), bank (of a river)," a word of obscure origin, chiefly Northern, which is probably from or related to dialectal German bräme "margin, border, fringe," from PIE *bhrem- "point, spike, edge." It was 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 perhaps akin to Old Norse barmr "rim, brim." "It became obs. in ME.; but was perhaps used by Spenser" [OED].

"to fill to the brim," 1610s, from brim (n.). The intransitive sense of "be full to the brim" is attested from 1818. To brim over "overflow" is from 1825. Related: Brimmed; brimming.