In Assassin's Creed Black Flag, Edward Kenway's pirate ship is called the Jackdaw.
Edward acquires the vessel early in the story and uses it to sail through the Caribbean. The Jackdaw functions as both transportation and a major part of gameplay: Edward can upgrade its hull, weapons, storage, and crew while taking on forts, rival ships, and naval contracts.
The ship's name connects Edward's pirate career with the game's broader themes of freedom, profit, and independence. Edward is not initially a committed Assassin; he becomes entangled with the Brotherhood and Templar conflict after impersonating another man and pursuing opportunities during the Golden Age of Piracy.
The Morrigan belongs to Shay Cormac in Assassin's Creed Rogue, and the Aquila belongs to Connor in Assassin's Creed III. Girty is associated with another character, so none of those names is Edward's ship.