Wordorigins.org

Gig is an interesting word with a variety of senses, not all etymologically related.

The oldest sense is that of a top or other whirling object. Originally whirligig, the origin is unknown but is probably echoic. From Promptorium Parvulorum Sive Clericorum, Lexicon Anglo-Latinum Princeps, c.1440:

Whyrlegyge, chyldys game, giraculum.
(Whirligig, child’s game, giraculum.)

This sense of top is the source of some other senses, such as a giddy or flighty person, fun, merriment, and a whim.