When asked about the scar on his lip, Jason always lied. It was easier that way. He thought it would be too embarrassing for the entire camp to know that the ever so powerful son of Jupiter had gotten his most visible scar from attempting to eat a stapler.
The most famous story was that he'd gotten it while fighting with a lion. But while the Hunters were visiting, Thalia loudly pointed out that there were no lions in San Francisco, and told everyone the truth.
Let's just say that Jason now has a life time supply of staplers, courtesy of two very obnoxious sons of Hermes.