Old fun thread.
The term "gig" with respect to an engagement for musicians first appeared in the journal "Melody Maker," in 1927 and 1927.
1926- One popular 'gig' band ....
1927- The seven-piece combination does many 'gigs' in SE London, but is hoping to secure a resident engagement in Leamington...
From the OED, 1987 supplement and in later complete editions.

Looks like the Brits are responsible. Is anyone familiar with the "Melody Maker"?