Random House, 1966 says,

1)light boat; light one-horse carriage; to ride in a gig [ME gigge, gig flighty girl,? gig top; cf. Norw giga to shake about

2) spearlike device from fizgigfisga harpoon; gig mill, a roller containing teasels, used for raising nap on a fabric

3) an official report of a minor infraction of regulations, as in school, the army, etc.

4) the term we all know from jazz

gigue (zh-long e g) a dance, jig, often forming the concluding movement in the classical suite [F, also < It. giga, orig., a fiddle or lute)

From a modern French dictionary, gigue is a haunch of venison; (coll) leg; (slang)long-legged gawky girl