early 14c., "grow, sprout, blossom," from Anglo-French burjuner, Old French borjoner "to bud, sprout," from borjon "a bud, shoot, pimple" (Modern French bourgeon), a word of uncertain origin. Perhaps from Vulgar Latin *burrionem (nominative *burrio), from Late Latin burra "flock of wool," itself of uncertain origin.
Some sources (Kitchin, Gamillscheg) say either the French word or the Vulgar Latin one is from Germanic (compare Old High German burjan "to raise, lift up"). The English verb is perhaps instead a native development from burjoin (n.) "a bud" (c. 1300), from Old French. According to OED, it died out by 18c. except as a technical term in gardening, and was revived early 19c. in poetry. Related: Burgeoned; burgeoning.