If Percy is thrown back in time, his powers could be enhanced. In the past, magic and the power of the gods were more prevalent. The very air might be charged with more divine energy. His ability to communicate with sea creatures could expand, as there could be species that are now extinct or have evolved in his own time. However, he might also face limitations. For example, if he goes back to a time when the gods were at war, some of his powers related to Poseidon might be restricted depending on his father's situation in that war.
One idea could be Percy going back in time to the era of the first Olympian gods and having to deal with the raw, unbridled power of those early deities. He might find himself in the middle of a power struggle between Zeus and his father, Kronos, before the Titanomachy. Another plot could involve Percy trying to prevent a major disaster in the past that has far - reaching consequences in his own time. For example, stopping the release of a powerful monster that was supposed to be sealed away long ago.
Percy might run into his old friends' ancestors. This could create some really strange and exciting interactions. For instance, if he meets an ancestor of Annabeth, he could learn things about her family history that she doesn't even know.
Well, Percy could end up changing the entire course of history in the 'Percy is thrown back in time' fanfiction. He might accidentally stop some important relationships from forming or start new ones that were never meant to be. Also, he could have a chance to see his loved ones who are no longer alive in his original time, which would be an emotional rollercoaster for him.
A third plotline could be Percy's search for a powerful artifact or knowledge in Tartarus that could help him in his future battles or save his friends. Along the way, he discovers secrets about the origin of the gods or the true nature of Tartarus itself, which changes his understanding of the world he thought he knew.
The Titans or Giants that were banished to Tartarus could be the main antagonists. They might see Percy as a threat or an opportunity to escape and seek revenge on the Olympians.
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.
Maybe a corrupted past version of a demigod. For example, a demigod who was promised great power by the Titans and turned against the Olympians. This demigod could see Percy as an obstacle to their plans and try to stop him at every turn. They would know the time period well and could use that to their advantage against Percy.