Perhaps in the fanfiction, Apollo kidnaps Percy because he thinks Percy has a connection to a prophecy that could either save or destroy the world of the gods. Apollo, being a god of prophecy himself, wants to control the outcome. So, he takes Percy away from his friends and allies to figure out how to use this connection to his advantage.
It could also be that Apollo is under the influence of some dark force. This force makes him kidnap Percy. Percy then has to use his wits and powers to not only escape but also to free Apollo from this influence. He might find allies among other minor gods or demigods who are also concerned about Apollo's strange behavior.
Percy would probably be angry at first. He has a strong sense of justice and being kidnapped against his will would make him fight back. He might immediately try to use his water powers to break free or at least create a distraction.
Well, perhaps Aphrodite kidnaps Percy because she wants to play matchmaker. She might see some potential love connection for him that he's been avoiding, so she takes matters into her own hands. Maybe she has a grand plan to set him up with a certain demigod or even a mortal, and she believes that by kidnapping him, she can force him to face his romantic destiny.
Well, it could be that Nico has some dark plan. Maybe he believes Percy is the key to saving someone he loves in the Underworld. He might think that by kidnapping Percy, he can force Percy to use his powers to bring that person back to life or something like that.
One possible plot could be Percy Jackson being chosen by Apollo to complete a special quest. Maybe he has to retrieve a lost artifact of Apollo that is crucial for maintaining the balance between the gods. Along the way, he might face various monsters and challenges, like fighting against the minions of a dark god who wants to use the artifact for evil purposes.
It could be about Percy having some sort of traumatic experience that leaves him scarred, and Apollo takes him in to raise him. Maybe Percy's experience was related to a battle or some sort of mythical encounter, and Apollo, seeing Percy's potential or just out of kindness, decides to be his guardian.
Perhaps the plot involves Percy helping Apollo with a secret mission. Since Percy is good at handling tough situations, he could be assisting Apollo in retrieving a lost artifact. They keep their friendship a secret because Apollo doesn't want the other gods to think he's relying too much on a demigod.
A plot could involve Naruto being sent to the Percy Jackson universe through some kind of magical accident. He meets Apollo, who is intrigued by Naruto's chakra and ninja techniques. They team up to stop an ancient evil that threatens both the ninja world and the world of the demigods. Along the way, Naruto has to adapt to the new rules and powers of the demigod world, while also teaching some of his ninja wisdom to Percy and his friends. There could be epic battles where Naruto's Rasengan combines with Apollo's solar powers to defeat powerful foes.
Well, one possibility could be that Percy, already a hero in his own right, finds himself in a situation where Apollo decides to adopt him. Maybe Percy has lost his connection to Poseidon for some reason, and Apollo sees his potential. Apollo could then train Percy in new ways of using his powers, not just related to water but also in areas like prophecy or archery, which are Apollo's domains.
Perhaps Percy kidnaps Annabeth because he has been misinformed about her. He might believe she has done something terrible, like betrayed the demigods or the gods themselves. As the story unfolds, he discovers the truth, that she has been framed, and then they have to work together to clear her name and stop the real wrongdoers.
Well, it could be about Percy and Apollo becoming friends in an unexpected way. Like, Apollo is in a situation where he loses his godly powers temporarily and Percy helps him out. During this time, they bond over their shared experiences of being different from regular humans. They start to understand each other's strengths and weaknesses, and form a strong friendship.