Jason has a "cute little scar on his upper lip" (3.6) we learn early on in The Lost Hero. Eventually we learn from his sister, Thalia, that he got the scar when "he tried to eat a stapler when he was two" (35.6).
At the beginning of the novel, Jason doesn't know how he got the scar because he has amnesia. But the scar is a sign that he does have a past—it proves he existed before, even if he doesn't remember. The scar is a mark of Jason's history and of his reality. It shows that he's here now and that he's been around for a bit.
The trick, though, is that Jason is not there. He's a character in a book, not a real person. His amnesia is actually the truth; he has no past before the story opens. The scar is a kind of mask—or a kind of Mist. The novel gives us the story about the stapler just as the Mist gives Piper the story that Jason is her boyfriend. Is the boyfriend story real? Is the stapler? Is the scar? The tiny cut is a quiet wound on the story, a place where the fiction hasn't quite closed over reality.