Gigs, are small wooden water crafts that have a coxswain and varied numbers of oars depending on the size of the boat.
Their purpose was to act as transport for crew of ships that were anchored outside of the harbor. They were designed for speed because the boat that got to the larger ships first, got the job. Hence, "It's your gig."
Merriam-Webster has this to say:
3a : a long light ship's boat
3b : a rowboat designed for speed rather than for work
Wikipedia on the (Cornish pilot) gig.
The Cornish pilot gig is a six-oared rowing boat, built of Cornish narrow leaf elm, 32 feet (9.8 m) long with a beam of 4 feet 10 inches (1.47 m). It is recognised as one of the first shore-based lifeboats that went to vessels in distress, with recorded rescues going back as far as the late 17th century. The original purpose of the Cornish pilot gig was as a general work boat, and the craft is used for taking pilots out to incoming vessels (see pilot boat) off the Atlantic. At the time, the gigs would race to get their pilot on board a vessel first (often those about to run aground on rocks) in order to get the job and hence the payment.