What? Why Wouldn't Stallone know the name of his daughter?
Wife is pregnant, daughter's first name is not yet decided.
John Spartan goes to cryo-prison for the rest of everybody's forseeable lives. He's basically dead.
Spartan's wife remarries to a "Mr Huxley". Spartan's daughter is then named "Lelina".
Spartan's wife/Lelina's mom dies in the earthquake. Lelina never learns of her Spartan connection.
Spartan asks Lelina about his family. The police computer says his wife died in the earthquake (Lelina: Yeah, that happened to lots of people, aka: Yeah, that happened to my mom too), but his daughter is still alive, and her identity is protected. Lelina would need to unseal the file to see any more than "wife (dead), daughter (alive)".
Edit: It's a ratburger situation. Spartan doesn't want to know if this theory is true or not, he just wants to eat his ratburger in blissful ignorance.
She was the kid at the beginning of the movie if I remember right.
That was a hostage being held for financial ransom.