There could be a fanfic where Percy is kidnapped by a cult that worships an ancient and evil god. They believe Percy has the key to awakening their god fully. They keep him in a secret lair filled with all kinds of magical traps and barriers. Or, perhaps a group of Titans who were thought to be defeated manage to come back and kidnap Percy as he is seen as a major threat to their potential return to power. Another scenario is that a powerful wizard from a different magical realm kidnaps Percy, thinking he can use Percy's unique demigod abilities to enhance his own magic.
Well, a plot could be that Percy is kidnapped by a sorcerer who has a vendetta against Poseidon. The sorcerer thinks that by taking Percy, he can get revenge on the sea god. Another could be that some jealous demigods from another camp, who feel overshadowed by Percy's fame, kidnap him. They might try to make him look bad in front of the other demigods. Also, it could be that a mysterious force from the Underworld kidnaps Percy because they fear his growing power and want to keep him out of the way.
One possible plot could be Percy getting a special power from Apollo's blessing. For example, he might gain the ability to heal faster, which would be really useful in his battles. He could use this new power to save his friends in a dangerous situation, like when they are facing a powerful monster and his friends are severely injured.
A really interesting plot could be that Sonic is kidnapped into another dimension by a powerful being. In this dimension, the laws of physics are different, and Sonic has to figure out how to use his speed in new ways to escape. His friends back in his own dimension are desperately trying to find a way to bring him back. Meanwhile, Sonic meets some strange new allies in this other dimension who help him in his quest to return home.
One possible plot could be Percy being stuck in a time loop where he has to keep saving Artimis from a recurring threat. Maybe a powerful Titan is constantly finding new ways to capture her, and Percy has to figure out the pattern within the loop to stop it for good. Another idea is that the time loop is a result of a curse placed on Artimis, and Percy has to team up with the hunters of Artimis to break the curse before the loop consumes them all.
There could also be a plot where the sirens are not the typical villains. In this fanfiction, they could be victims of a greater evil in the ocean. Percy has to team up with them to fight this common foe. It would be a great twist on the traditional view of sirens in the Percy Jackson universe.
One possible plot could be that the kidnappers are actually fans who have gone too far in their obsession. They take Tom Holland, thinking they can have him all to themselves. Another could be that he's kidnapped as part of a ransom plot by a criminal group who wants a large sum of money from his management. Maybe it's also a case where a supervillain from one of his movie universes kidnaps him for some strange, other - worldly reason.
One possible plot could be that Charlie is kidnapped by a group who wants to use his mathematical genius for their own illegal purposes, like cracking codes for a major heist. Another could be that a deranged fan kidnaps him because they believe Charlie has some sort of secret knowledge that can change the world. Maybe it's a rival mathematician who is jealous of Charlie's fame and kidnaps him to try and steal his research ideas.
One interesting plot is Percy being summoned by a time - traveling device. He is brought to a future where the Greek gods are forgotten and the world is in chaos due to the lack of their influence. Percy has to find a way to bring back the knowledge of the gods and restore order, all while dealing with the strange new technology and a society that has lost its moral compass.
Perhaps Octavian's sickness is a way for the author to explore themes of forgiveness and acceptance. He could be a really unlikable character at first, but as he gets sick, he starts to change. The other characters see this change and start to question their own prejudices against him. His sickness could be a catalyst for him to make amends with those he has wronged in the past. And the story could end with him either fully recovered and a changed person, or with his death being a turning point for the other characters to become better.
In some fanfictions, Typhon is not just a mindless monster. He might have motives and desires. For instance, he could be seeking revenge against the gods for imprisoning him. This makes him a more complex character and gives the story more depth. Writers can then explore the moral implications of whether the demigods should just destroy him or try to reason with him.
It could be Luke Castellan. He has a history of being against Percy and the Olympians. He might kidnap Percy to gain an advantage in his own quest for power. He could use Percy as a bargaining chip with the Titans or other powerful forces.