Perhaps they meet a group of outcasts who have their own unique abilities. These outcasts could teach Percy and Annabeth new fighting techniques or ways to use their powers more effectively. They might join forces with them to take on a common enemy that is threatening both the outcasts and the world they come from.
They might embark on a journey to find a mysterious mentor. This mentor could be an ancient being who has knowledge of a powerful weapon or a way to defeat a great evil. Along the way, they have to face various challenges like crossing a treacherous river filled with magical beasts or solving puzzles in an abandoned temple.
His friends at the camp would be worried. They might split up to search for him, which could put them in danger too. And this could lead to a rift in the group dynamics at the camp, with some blaming Percy for causing chaos.
One reason could be to avoid a dangerous prophecy that they believe the camp can't protect them from. In some fanfictions, they might think that by running away, they can face the threat on their own terms.
It could be Chiron, the centaur. He's always keeping an eye on the campers and would likely be concerned if he caught them up to something. He might give them a stern talking - to and some extra chores as punishment.
There could be a secret passageway in the school that leads to an ancient mythological place. Percy and Annabeth discover it by accident. Percy is excited to explore but Annabeth warns him to be cautious. They enter together, facing various traps and puzzles inside. Along the way, they might meet some minor gods or spirits who are either helpful or try to hinder their progress. This adventure would test their skills and teamwork, just like their quests in the original stories.
They might choose to run away because they want privacy. After all the battles and saving the world multiple times, they just want some time alone without any interruptions from other demigods or the gods. So running away seems like a good option.
Perhaps he was influenced by some new friends he met who had a different lifestyle or view of the world. They might have convinced him that running away from home was an exciting adventure and a way to break free from the restrictions at home.
It could be that Annabeth has a misunderstanding about Percy. She might think he's a threat to her position or status at the camp, so she bullies him to try and keep him in check.
It could bring them closer. Percy might find it endearing that Annabeth likes fanfiction about them, and it could lead to more open conversations between them about how they are perceived by fans.
In the fanfiction, they could also be on a quest to retrieve a powerful ancient Greek artifact. This artifact could be something like the Helm of Darkness that has been lost for centuries. Along the way, they would face many challenges. They would have to interact with the gods and goddesses of Ancient Greece, who might not all be friendly. Percy and Annabeth would rely on their friendship and unique skills to overcome all the obstacles in their path.
They might face new and strange monsters. For example, they could encounter a sea - serpent that has the power to control the weather around the Argo 2. Percy would use his water - controlling abilities to try and fend it off while Annabeth comes up with a strategic plan to defeat it.