Saul

If you have Netflix check out the opening scene in season 1, episode 4 and you kind of see the origin of the name. It's a flashback scene to when he was running scams with a friend of his and he's walking down the street with the poor sap he's working and they have the following conversation:

Poor sap: "hey bro, uh I never did catch your name?"
Jimmy: "Saul."
Poor sap: "Saul?"
Jimmy: "S'all good man.....get it?"