Sorry guys, I skipped half the article from my dictionary about the "gigot" (mutton leg): "is generally considered as derivating (end 14th century) from the Old French guigue (c. 1120 - 1150) "3 string instrument of music". Guigue was borrowed from the Old High German Giga (12th century) "string instrument". But this is related to "gigue" = "jig" and not to "gig".

I don't think that it'll help much but the Oxford concise dictionary of English Etymology, (1996) says about "gig":
† flighty girl (XIII)
† whipping-top (XV)
† fancy, whim (XVI)
† (dial.) fun, glee; (dial.) odd person, fool (XVIII)
Light, two-wheeled carriage; light ship's boat (XVIII)
All these uses may be referred to the general notion of light or quick movement, which is also that of the later JIG; but the history of both words is obscure.