'budge' originates from Middle English (records from the 14th–15th century), possibly influenced by Anglo-French words such as 'bouge'/'buge' referring to a pouch or hide, where the sense was associated with leather or pelts.

'budge' first appeared as a noun meaning 'a sheepskin or lamb's fur' in Middle English; from that noun sense a verbal sense 'to move or give way' developed in later English, and the figurative sense 'to yield' followed.

Initially, it referred to a type of hide or fur; over time the primary modern meanings shifted to physical movement ('move slightly') and figurative yielding ('give in').