"small open vessel (smaller than a ship) used to cross waters, propelled by oars, a sail, or (later) an engine," Old English bat, from Proto-Germanic *bait- (source also of Old Norse batr, Dutch boot, German Boot), possibly from PIE root *bheid- "to split," if the notion is of making a boat by hollowing out a tree trunk or from split planking. Or it may be an extension of the name for some part of a ship.
French bateau "boat" is from Old English or Norse. Spanish batel, Italian battello, Medieval Latin batellus likewise probably are from Germanic. Of serving vessels resembling a boat, by 1680s. The image of being in the same boat "subject to similar challenges and difficulties" is by 1580s; to rock the boat "disturb stability" is from 1914.
for historical evolution, see V. Used punningly for you by 1588 ["Love's Labour's Lost," V.i.60], not long after the pronunciation shift that made the vowel a homonym of the pronoun. As a simple shorthand (without intentional word-play), it is recorded from 1862. Common in business abbreviations since 1923 (such as U-Haul, attested from 1951).
The substitution of Middle English -o- for Old English -u- before -m-, -n-, or -r- was a French scribal habit before minims to avoid misreading the letters in the old style handwriting, which jammed them together. The practice transformed some, come, monk, tongue, worm.
It forms all or part of: abbreviate; abbreviation; abridge; amphibrach; brace; bracelet; brachio-; brachiopod; brachiosaurus; brachy-; brassiere; breviary; brevity; brief; brumal; brume; embrace; merry; mirth; pretzel; vambrace.
It is the hypothetical source of/evidence for its existence is provided by: Greek brakhys "short;" Latin brevis "short, low, little, shallow;" Old Church Slavonic bruzeja "shallow places, shoals;" Gothic gamaurgjan "to shorten."