An 1894 Slang Dictionary gives one meaning of "gig" as fun, frolic, a spree . It quotes the following example from 1820:

In search of lark or some delicious gig,
The maind delights on, when 'tis in prime twig."

It claims this sense derives from an Old French sense ef gigue meaning a jig, a romp.

Seems to me this is not far from our modern sense.

Regards
p.s. Mind you.... The 1811 Dictionary of the Vulgar Tongue glosses "gig" as "a woman's privities" !