Since there is a French word "gigue" with a similar meaning, I lean toward the theory that the currently familiar usage of "gig" probably dates back to the earliest days of jazz in New Orleans.

Many of the first generation of jazz musicians were French-speaking and bilingual Creole "gens de couleur" (people of color). Seems to me that these would have been the guys to first use and popularize the term "gig" for musical job or engagement.

(Incidentally, these Creoles were much more likely to have received formal education, including musical education, than their English-speaking black and white contemporaries, and so had great influence in the formative years of jazz as bandleaders, etc.)