"light, frivolous woman," 1520s, first element of uncertain origin, second element is Middle English gig "frivolous person" (see gig (n.1)).
Entries linking to fizgig
gig n.1
"light, two-wheeled carriage, usually drawn by one horse" (1791), also "small boat," 1790, perhaps imitative of bouncing. There was a 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." Similar to words in continental Germanic for "fiddle" (such as German Geige); the connecting sense might be "rapid or whirling motion."