He could potentially change some events in the past that had a big impact on his present life. Say, he could prevent some of the early attacks by monsters that made his life so difficult growing up. This would completely alter the course of his life and the lives of those around him. It's a really cool concept to think about how one small change in the past could have a huge ripple effect in the future.
In such a fanfiction, Percy might use his knowledge of the future to change some key events in the past. For example, he could prevent some of the earlier disasters or conflicts in the Percy Jackson universe. Maybe he'd try to stop the first encounter with a certain monster in a different way to avoid the chaos it caused.
He could meet his parents when they were young. Maybe he'd get to see his mother before she met his father and understand more about her life back then.
In the 'Percy goes back in time fanfic', he may also meet his younger self. This could lead to some really interesting and confusing situations. He would have to be careful not to disrupt the timeline too much while still achieving whatever his goal is for going back in time.
In fanfiction, if Percy Jackson goes to highschool, he might face challenges like hiding his demigod powers from his classmates. He could also encounter new mythical creatures that are drawn to the school environment.
In a fanfiction where Percy Jackson goes to Azeroth, he could form alliances. Since he's used to being in a team in his own world, he'd likely team up with some of the heroes in Azeroth. For example, he could join forces with Thrall and help him in his quest for the Horde. His sword - fighting skills from Camp Half - Blood would come in handy in battles against the Burning Legion or other enemies.
If Percy Jackson goes to UA, there would be a huge culture clash at first. The UA students are used to a very structured hero training system, while Percy is more of a 'go with the flow' kind of demigod. But over time, he could teach them about improvisation in battle and they could teach him about teamwork in a more organized way. He might also introduce the concept of Greek mythology to the UA world, which could open up new ways of thinking about powers and abilities.
Well, he might get into all sorts of adventures. He could meet the dragons in Alagaesia and try to use his water powers to his advantage against them. Maybe he'd team up with the local heroes there and learn new combat styles.
Perhaps Percy would see a future where the monsters have evolved in strange ways. They could be more intelligent and organized, posing an even greater threat to the demigods. He might also find that some of his old friends have made great sacrifices and that he has a new role to play in this future world, like being a mentor to the new generation of demigods.
In this fanfiction, Percy might decide to break away from Camp Half - Blood. He could start exploring the mortal world on his own terms, using his demigod powers in ways that the camp rules don't allow. Maybe he teams up with some unexpected allies, like a group of outcast demigods or even some friendly monsters.
In this fanfiction, perhaps Percy will make new friends at Goode who end up getting involved in his demigod world. There could be a plot where the school is in danger, and Percy has to use his powers and knowledge of the mythological world to protect it. Maybe he'll also have some interesting interactions with his teachers who suspect there's something different about him.