One idea could be exploring Peter's life before he became the famous Peter Pan. Maybe he had a best friend in the orphanage who was left behind when he went to Neverland. Another plot could be about the origin of the starstuff and how it affects different characters in ways not shown in the original. For example, a minor character could gain some special powers from a small amount of starstuff and use it for good or bad.
One underrated character is Prentiss. He's often just in the background in the original, but in fanfiction, he could be the hero. Maybe he discovers a hidden power within himself that helps Peter and the gang. Another is Ted. He could be given more spotlight by having his own backstory explored, like how he ended up on the ship in the first place.
Perhaps there's been some sort of miscommunication or misunderstanding among Peter and his friends. Maybe a villain has manipulated the situation in such a way that his friends now think Peter has betrayed them, so they've cut off all ties with him. This would create a very lonely and difficult situation for Peter in the fanfiction.
Peter could be just an ordinary person with an extraordinary love for dogs. He might be a local boy who has always been fascinated by the tower and decides to bring his furry friends along on his adventure.
Peter could be a disgruntled noble in the fanfiction. He might be someone who has been overlooked for promotions or honors by the emperor, and this has led to his resentment and desire to kill the emperor.
Again, without more context, it's hard to say exactly who Peter Braestrup is in this 'big story'. He could be an author, an activist, or someone who made a mark in a particular industry.
The story behind 'Peter Peter Pumpkin Eater' is kind of mysterious. Nobody really knows for sure how it came to be. It's just one of those rhymes that has always been there, without a detailed explanation of its origins.