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.