One possible plot could be that when Percy wakes up after centuries, he finds the world he knew completely changed. Maybe all his friends are long gone and he has to deal with a new set of demigods who see him as a relic from the past. He might struggle to adapt to the new technology and social structures.
Well, it could also be that Percy was put to sleep as part of a grand plan by some ancient force. When he wakes up after centuries, he discovers that he is the key to solving a great threat that has emerged in the modern world. He has to quickly learn about the new enemies and allies, and his long sleep might have given him some sort of enhanced abilities or knowledge that he needs to use to save the day.
It could be a new god or goddess who has risen to power during Percy's long sleep and sees him as a threat to their rule. This new deity might be jealous of Percy's past heroics and fame among the demigods and wants to eliminate him.
One possibility is that they've had a long and tiring adventure. After facing numerous challenges, they find a moment of peace and Annabeth, exhausted, falls asleep on Percy. It could also be a sign of their deep trust in each other. In fanfiction, this might be used to show the closeness of their relationship.
One possible plot could be Percy's struggle to survive in a new, unknown realm after being banished. He might have to find new allies and face new enemies.
One possible plot could be Percy struggling to control new, strange powers that come with the curse. Maybe he has to constantly battle against the dark impulses the curse brings out in him.
In a more complex plot, Percy could be forced into a boxing match by a powerful enemy who wants to humiliate him. But Percy turns it into an opportunity to unite the demigods. He uses the boxing ring as a platform to give inspiring speeches between rounds. And as the matches progress, he gradually gains the respect of both his allies and former foes through his tenacity and skill in the ring.
Perhaps in the fanfiction, Nico has been tricked by an enemy. This enemy convinces Nico that Percy is a threat to their world. So, Nico decides to take matters into his own hands. Another possibility is that Nico's own internal demons, related to his past traumas and his complex relationship with the gods, drive him to see Percy as an obstacle that needs to be removed.
Perhaps Percy has received some bad news or a prophecy that makes him think that by ignoring everyone, he can protect them. So he distances himself from his friends and family, and the fanfiction could explore how they react to his sudden cold - shoulder. They might try to figure out what's wrong with him and how to bring him back to his normal self.
In the 'percy is turned into a monster fanfiction', it could start with Percy being tricked by a powerful deity into becoming a monster. His transformation might be a way for the deity to use Percy as a weapon against other gods or demigods. Percy, in his monster form, has to fight against this control and find a way to break free. He might encounter other characters who help him understand his new form and use it to his advantage rather than being a slave to the deity's will. Along the way, he discovers new aspects of his identity and the world around him that he never knew before.
One possible plot could be that Percy is found by a kind-hearted hermit who lives in the woods. The hermit raises Percy in seclusion, teaching him basic survival skills and a unique form of magic unknown to the outside world. As Percy grows, he starts to have strange dreams about his past and his real parents. Eventually, he decides to leave the safety of the woods to search for his true identity.
One possibility is that there is a prophecy that involves Percy and the hunters in a way that the hunters misinterpret. They believe that by torturing Percy, they can prevent some great disaster. However, Percy is actually an important part of the solution, not the problem. As the story progresses, he could start to communicate with some of the more sympathetic hunters and try to make them see the truth.