The meaning progressed from "compliant, obliging," through "lively, jolly," "healthily plump, vigorous and attractive," to (in women, and perhaps influenced by lusty) "attractively plump, comely" (1580s). In Johnson  the primary meaning still is "obedient, obsequious." It was used especially of women's figures from at least 1870s, and by 1950s it had begun to be used more narrowly for "bosomy" and could be paired with slim (adj.). Among its cognates are Dutch buigzaam, German biegsam "flexible, pliable," which hew closer to the original English sense.
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.