Old English brad "wide, not narrow," also "flat, open, extended," from Proto-Germanic *braidi- (source also of Old Frisian bred, Old Norse breiðr, Dutch breed, German breit, Gothic brouþs), which is of unknown origin. Not found outside Germanic languages. There is no clear distinction in sense from wide. Of day or daylight, late 14c.; of speech or accents, 1530s. Related: Broadly; broadness.
"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].
<a href="https://www.etymonline.com/word/broad-brim">Etymology of broad-brim by etymonline</a>
Harper, D. (n.d.). Etymology of broad-brim. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/broad-brim