"light carriage, small boat," 1790, perhaps, on notion of bouncing, from Middle English ghyg "spinning top" (in whyrlegyg, mid-15c.), also "giddy girl" (early 13c., also giglet), from Old Norse geiga "turn sideways," or Danish gig "spinning top."
"job," first used by jazz musicians, attested from 1915 but said to have been in use c.1905; of uncertain origin. As a verb, by 1939. Related: Gigged; gigging.