Spoilers for The Punisher Season 1 ahead. When we meet back up with the certainly-not-dead Frank Castle in Marvel's The Punisher on Netflix, he's lying low in a couple of ways. He doesn't speak much, he's got a construction job, and he's going by a new name. Pete Castiglione is Frank Castle's alias on The Punisher and it's nice nod to Marvel lore.
In the comics, Francis Castiglione was Frank Castle's given name, and he changed it to illegally enlist in a third military tour during the Vietnam War. So there are a lot of characters named Castiglione in Marvel comics: Frank's wife Maria, father Mario, mother Louisa, uncles Rocco and Fredo, and even his grandmother Esmerelda. There's no "Pete" Castiglione in Marvel comics, but the surname is a clear nod to this part of The Punisher's origin story.
Why Peter, then? That isn't really explained in the series. While The Punisher does sometimes share storylines with Peter Parker in Marvel comics, and they both live in Queens, that would be an odd reference to make in the Netflix series, especially after Civil War and Spider-Man: Homecoming officially placed the two characters in the same cinematic universe. Maybe the Punisher's alias is paying homage to a fallen friend or family member. Maybe it's a nod to someone in the real world.
Maybe it's just a short name that he thought would be easy to remember. That's usually a good strategy for picking a new identity — choose something that sounds like your own name, or has the same number of syllables, or whatever it takes to help you remember the lie you're telling.
Frank not only uses that name at work in the first episode, but he continues to introduce himself as Pete throughout the season. When he starts to get close to Micro's family, he's known as Pete. He doesn't create too much of a backstory for himself, choosing instead to be a strong and silent bearded "hipster" who haunts diners. That can't last forever, obviously, as the story goes on and more and more people realize that Frank Castle is alive he will be forced to drop the act.
Still, it's cool that The Punisher is so hardcore, he needs two secret identities. Right? That's one way of looking at it. In theory, "Pete Castiglione" is a way for Frank to move on from his past traumas. It doesn't seem to be working in The Punisher Season 1, but it's an attempt. He can be someone else. In an interview with The AV Club, actor Jon Bernthal says he never thinks of the character as The Punisher first.
"He’s a father and husband who had [his family] stripped from him. As a guy who’s a young father and a husband, before I’m anything else in this world, that’s number one for me. I love my family with everything I have, and just to imagine what that would be like for them to be taken from me, that starts a spiral of fear and insanity and rage in me that takes me to some really scary places. And it’s always been my theory in my career that if something scares you, you run right toward it. And that’s what Frank Castle has been for me."
In this new series after the events of Daredevil Season 2, Frank no longer needs to be defined by what happened to him and he can start a new life. He's still living with that and recovering from it, but that's what "Pete" should represent — the chance to have a future. If Frank Castle's past ever stopped haunting him, maybe it could.