I don't know for sure as there are many possible characters in Twilight fanfics. It could be Bella and Edward, where Bella has some new power or quality that has Edward wrapped around her little finger.
Since Twilight has a large cast, it could also be that it focuses on secondary characters like Leah Clearwater. Maybe she discovers a new ability or trait that makes other werewolves or even some vampires in the story wrapped around her little finger. Without reading the fanfic, it's hard to be certain about the exact main characters.
It might be a new character introduced in the fanfic. For instance, a young vampire who is extremely manipulative and has some of the main Twilight characters like Jacob or Carlisle wrapped around her finger. Or perhaps it could be Rosalie with some new plot twist that makes other characters follow her every whim within the context of this fan - created story.