I was bemused and a little amused to find an entry in the OED for a word whose meaning is unknown. Seems to confound the point of a dictionary but I admire their completism. They have a few cites and an etymological note.
† gig, v.1
Pronunciation: Brit. /ɡɪɡ/, U.S. /ɡɪɡ/
Etymology: < gig n.1 (sense 1).
The verb seems literally to denote the action of some kind of ‘gig’ or whipping-top of peculiar construction, having inside it a smaller ‘gig’ of the same shape, which was thrown out by the effect of rapid rotation. Hence to gig (out) appears to be used fig. with the sense ‘to throw out or give rise to (a smaller repetition of itself)’. The Dicts., on the ground of the Dryden quot., have plausibly, but erroneously, explained the transitive vb. as meaning ‘to engender,’ assigning to it a derivation from Latin gignĕre.
a. intr. (Sense obscure: see etymological note.)
1651 J. Cleveland Poems 44 No wonder they’l confesse no losse of men; For Rupert knocks ‘em, till they gig agen.
1690 Dryden Amphitryon Prol. sig. Aiv, Yet in Lampoons, you Libel one another. The first produces still, a second Jig; You whip em out, like School-boys, till they gig: And, with the same success..For, ev’ry one, still dwindles to a less
b. trans. (Sense obscure: see etymological note.)
1659 T. Burton Diary (1828) IV. 185 One question gigs out another. We shall never end.
1677 J. Lake & S. Drake in J. Cleveland Clievelandi Vindiciæ Ep. Ded. sig. A4, How many of their slight productions may be gigged out of one of his pregnant Words?
1690 Dryden Amphitryon iii. 27 Sosia. You, my Lord Amphitryon, may have brought forth another You my Lord Amphitryon..and our Diamonds may have procreated these Diamonds… Phædra. If this be true, I hope my Goblet has gigg’d another Golden Goblet.
(I was also surprised the OED doesn’t yet have gig meaning gigabyte, nor gigabyte, but I suppose they’ll get to it.)