One idea could be that Percy goes back to the time when the gods first came to power. He could try to change the way the gods interact with demigods from the start, leading to a whole new world order for demigods. Maybe he'd teach the early demigods some of the skills he has, like advanced sword fighting techniques.
One interesting plot idea could be Percy going back in time to the era of the Greek heroes. He could meet Hercules and learn from him. Maybe he would have to face the same challenges Hercules did, but with his own modern - day skills and knowledge. Another idea is that he goes back to the time when the gods first emerged and gets involved in the power struggles among them. This could change the entire course of mythology as we know it.
One idea could be that the demigods are sent back to prevent a great disaster that was actually caused by one of their own kind in the past. They have to figure out who it was and stop it without changing too much of the future.
Jon Snow could be sent back to a time just before Ned Stark's father was killed. With his future - learned wisdom, he could try to save his grandfather and thus change the entire Stark family's fate. He may also discover secrets about his own lineage that were lost in time. This could lead to him having more power and influence in the past, as he might be able to claim certain rights or titles based on this new - found knowledge.
A possible plot is that Anakin is sent back in time and ends up meeting his mother again when she was younger. This encounter could give him a chance to save her from the Tusken Raiders, which might then have a profound impact on his emotional state and his path in the Force. He could also use this opportunity to train himself differently, perhaps learning from the Jedi of that era in a more positive way, without the strictures that later led to his disillusionment.
One idea could be Percy going back in time to the era of the Greek gods' creation. He could witness the birth of the first demigods and learn the true origins of his powers. Another plot could involve him traveling back to a time when a great prophecy was first made and trying to change the course of events to prevent a future disaster.
Well, perhaps a major plot twist is when Percy goes back in time and realizes that the events he thought were accidents in his own time were actually carefully orchestrated by a character from the past. This would add a whole new layer of mystery and conspiracy. Also, he might find that a prophecy he was trying to prevent in his time was actually meant to happen in the past for a greater good, which would be a huge turn for his mission.
In a 'Percy back in time fanfic', Percy could travel back to the time of the Titans' rule. He would be in a world full of chaos and oppression. Maybe he teams up with some of the early heroes who were fighting against the Titans secretly. He could learn new powers or fighting techniques from that era. His presence there could also cause some of the Titans to be more cautious, which could potentially change the outcome of the Titan - god war. This would then have a huge impact on the present that he came from, as the world would be very different in terms of power dynamics and relationships between different mythological beings.
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.
A possible plot is that Bellatrix is sent back to the time when the Goblin Rebellions were happening. She could side with the goblins to gain their trust and then double - cross them to steal their most precious treasures. Another idea is that she is sent back to the time when the Ministry of Magic was first established. Bellatrix might try to infiltrate it from the start and plant the seeds of dark magic within its very foundation. Also, she could be sent back to a time when a famous magical beast was first discovered, and she tries to capture and control it for her own evil purposes.
One interesting idea could be Percy's struggle to survive in Tartarus. He might have to face new and terrifying monsters there. Another idea could be his journey of self - discovery in such a harsh environment. Maybe he'll find new powers or a deeper understanding of his own capabilities.