I believe it might possibly Jack Teague as his father, introduced in Pirates of the Caribbean: At World's End and seen in Pirates of the Caribbean: Stranger Tides, is named Edward Teague
Edward Teague was an infamous pirate captain in the Caribbean and father of Jack Sparrow. A legendary pirate in his own right, Teague occasionally re-appeared in the life of his son Jack, who followed in his buccaneering footsteps. Teague was an outlaw, but with an aristocratic bearing; a fearsome figure, but with unmistakable touches of sweetness and vulnerability.
At some point in his career, Captain Teague became part of the Brethren Court, obtaining the title of Pirate Lord of Madagascar, and later Keeper of the Code. His candor, rigorous personal integrity, and uncompromising commitment to the Pirate Code had earned him the fear and respect not only of the other Pirate Lords, but of his son Jack as well. Although he appeared as a "stay at home" pirate, Teague would still be active in his later years. Teague would take part in the Brethren's stand against Cutler Beckett's Armada in the War Against Piracy, where they would choose the next Pirate King; and followed Jack in his search for the fabled Fountain of Youth, where he would warn his son of the perilous voyage.
Source: Edward Teague
Now we know that Teague is Captain Teague's last name as in the Trivia section on the same page
There was some controversy towards Teague's first name, as he was only referred to as "Captain Teague" in the POTC films. In the At World's End video game, while playing as him fighting enemies, Teague can be heard saying: "That's what you get for challenging Captain Edward Teague". The name was designated in the novel The Price of Freedom, though any first name has yet to be mentioned in the films.
So while it's not confirmed in the films that his first name is Edward it does confirm that Teague is his last name. since generally children receive their father's last name's like Bill Turner -> Will Turner and Weatherby Swann -> Elizabeth Swann logically this would mean Jack's last name would come from his father and thus be Jack Teague.
