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.
His friends from his normal time period could also be important. They might be the reason he decides to go back in time in the first place, like to save one of them from a past danger.
One possible plot could be that female Percy goes back to the time when the first big prophecy was made. She uses her knowledge from the future to try and change things for the better. Maybe she tries to prevent some of the early tragedies that happened to her friends. She could also try to change the way the gods interacted with demigods in the past, knowing how it all turned out in her original time.
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.
Another possibility is a demigod from the past who is jealous of Percy's knowledge and power. This demigod could be someone who was once in a position of power or influence but feels threatened by Percy's sudden appearance and her attempts to change things. They might try to undermine her at every turn, whether by spreading false rumors about her or directly attacking her when she least expects it.
He might meet Luke before he turned bad. Percy could try to change Luke's fate, convince him not to follow Kronos. It would be a great chance to save a friend and prevent a lot of the chaos that happened later.
There are many fanfics that play with the idea of time travel. For Bella going back in time to meet Jasper, the story could explore how their prior knowledge of each other in the future affects their initial meeting in the past. For example, does Bella already trust Jasper because she knows him from the future? This could create a really unique dynamic in the story.
One possible plot could be Anakin going back to a time before his fall to the dark side. He could use his knowledge of the future to prevent certain tragedies. For example, he might stop the Clone Wars from starting, thus saving countless lives. This would also give him a chance to redeem himself and perhaps even change the fate of the Jedi Order.
Another plot could be that Bella is sent back in time by some magical force related to Elijah. When she arrives, she finds Elijah in a difficult situation, like being trapped by an enemy. Using her wits and any powers she might have, she helps him escape. This starts a chain of events where they become allies and then, over time, more than that. They might travel through different historical events together, with Bella always having the knowledge of the future in the back of her mind to help them out.
One potential plot could be Bloom going back to the time when the first great war in the magical world happened. She could try to stop the war from starting, which would change the entire history of the magical world. This would also mean that the relationships between different magical species would be completely different in the present.