Perhaps Percy is sent to the mental hospital as a form of punishment by the gods. While there, he starts to see hallucinations of his past battles and friends. But he soon realizes that these are not just hallucinations but some sort of message. With the help of a mysterious patient who has some special knowledge of the mythological realm, Percy has to find a way to redeem himself and leave the hospital before it's too late.
One possible plot could be that Percy wakes up in the mental hospital with no memory of how he got there. He starts to notice strange things like the nurses and doctors seem to have an otherworldly aura. As he tries to regain his memory, he discovers that the hospital is a front for some sort of mythological experiment. He then teams up with other demigods who are also trapped there to escape and expose the truth.
It's possible that the main antagonist is a new mythological creature that has the ability to manipulate minds. It has lured Percy and other demigods to the mental hospital and is slowly driving them insane in order to feed off their fear and chaos energy.
Well, he might meet other characters who also claim to have had strange experiences. They could form a sort of alliance, thinking they are all in the same boat, not realizing Percy's experiences are real in the context of the Percy Jackson universe. They could plan an escape together, believing they are being wrongly held.
One idea could be Percy starts having visions related to his demigod adventures in the mental hospital. The staff think it's part of his mental illness, but it's actually his connection to the Greek world trying to reach him. Maybe he meets other patients who are actually demigods in hiding too, and they have to figure out how to use their powers subtly without getting caught.
Another possibility is that Percy has suffered some sort of mental trauma from his battles. He's having nightmares and flashbacks, and his friends think the mental hospital can help him. But once inside, he realizes that there are dark forces at work in the hospital. He has to use his demigod powers subtly to protect himself and the other patients from these unseen threats.
Perhaps Sakura is in the mental hospital because she was wrongly committed. In the fanfiction, she could be trying to prove her sanity. Along the way, she forms unexpected friendships with other patients and staff members. These relationships help her on her journey to regain her freedom and also help her come to terms with parts of herself she didn't understand before.
Perhaps a powerful monster in disguise as a doctor or staff member. They could be using the hospital's resources to create a portal to the underworld or something equally dangerous, and Percy has to stop them. They would be constantly trying to drive Percy crazy or keep him from interfering with their plans.
One idea could be that Percy starts seeing strange things in the mental hospital that are actually manifestations of Greek monsters from his past. Maybe he has to convince the doctors and other patients that he's not crazy while secretly fighting off these threats.
One possibility is that Percy sacrifices himself to save his friends and Camp Half - Blood from a powerful, unbeatable enemy like an ancient primordial god. He might use his powers to create a shield or diversion that allows everyone else to escape while he faces certain death.
One possible plot could be that Percy is overwhelmed by the pressure of his demigod duties. He has constantly faced life - threatening situations, seen friends die, and felt the weight of the prophecy on his shoulders. In this fanfic, he might see shooting himself as a way to end the cycle of pain and danger for himself and those he loves.
Annabeth could be in the mental hospital as a patient who is overly logical and her intelligence is misconstrued as obsessive - compulsive disorder. She tries to analyze everything in the hospital, from the routines of the staff to the behavior of the other patients. Grover could be there too, his fear and nervousness around humans being seen as a social anxiety disorder. Percy, as always, is the heart of the story, trying to figure out how to use his powers to protect his friends and also deal with his own 'diagnosed' problems.