Here's a link with another suggestion, taken from the Random House Historical Dictionary of American Slang.

The udea is that it's linked with the menbaing of "gig" for a carriage, and that this comes from an earliuer meaning in whch a "gig" was used to mean something spinning, as in "whirligig". So the suggestion would be that it was applied to parties and dances, analogous to the expression "social whirl".

So you get to play at place where they are having a gig.

Well, it's as good an expalnation as any.