1580s (intransitive) "to move, stir, change position, give way a little;" 1590s (transitive) "change the position of;" from French bougier "to move, stir" (Modern French bouger), from Vulgar Latin *bullicare "to bubble, boil" (hence, "to be in motion"), from Latin bullire "to boil" (see boil (v.)). Compare Spanish bullir "to move about, bustle;" Portuguese bulir "to move a thing from its place." In 16c. canting slang, "a general verb of action, usually stealthy action" (Farmer, "Musa Pedestris," who gives among his examples budge a beak "to give the constable the slip," budge out or off "to sneak off"). Related: Budged; budging.
Old English bycgan (past tense bohte) "get by paying for, acquire the possession of in exchange for something of like value; redeem, ransom; procure; get done," from Proto-Germanic *bugjan (source also of Old Saxon buggjan, Old Norse byggja, Gothic bugjan), which is of unknown origin and not found outside Germanic.
The surviving spelling is southwest England dialect; the word was generally pronounced in Old English and Middle English with a -dg- sound as "budge," or "bidge." Meaning "believe, accept as true" is attested by 1926. Related: Bought; buying. To buy time "prevent further deterioration but make no improvement" is attested from 1946.
early 15c., bouget, "leather pouch, small bag or sack," from Old French bougette, diminutive of bouge "leather bag, wallet, pouch," from Latin bulga "leather bag," a word of Gaulish origin (compare Old Irish bolg "bag," Breton bolc'h "flax pod"), from PIE *bhelgh- "to swell," extended form of root *bhel- (2) "to blow, swell."
The modern financial meaning "statement of probable expenditures and revenues" (1733) is from the notion of the treasury minister keeping his fiscal plans in a wallet. Also used from late 16c. in a general sense of "a stock, store, or collection of miscellaneous items," which led to 18c. transferred sense "bundle of news," hence the use of the word as the title of some newspapers.