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.
Maybe Percy uses his time in the mental hospital to learn more about himself on a deeper level. Since he's away from the chaos of his normal demigod life, he has time to reflect. He could discover new aspects of his powers or even find a way to better control his ADHD and dyslexia, which are related to his demigod nature. And all this while trying to convince the hospital that he's not crazy.
In the mental hospital, Percy could be subjected to various tests. These tests might accidentally trigger some of his demigod abilities. For example, if they were to test his reflexes, his super - fast demigod reflexes could be exposed. This could lead to the hospital staff either thinking he's even more of a danger or starting to suspect that there's something truly extraordinary about him.
In fanfiction, Percy could be in the hospital because he was captured by an enemy and tortured. Now he's there being healed by the camp's healers or some advanced demigod medical technology. Another possibility is that he's having some sort of identity crisis related to his demigod status and it's causing him physical and mental stress, so he's in the hospital for both medical and psychological help.
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 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 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.
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.
In the 'Percy Jackson Disappears' fanfiction, perhaps Percy could be kidnapped by an unknown enemy. Maybe it's a new villain who has a vendetta against the demigods. This would lead to his friends, like Annabeth, going on a desperate search for him, facing many dangerous challenges along the way.
Well, in this fanfiction, Percy's immortality could lead to some really interesting character development. He may start to feel lonely as he realizes that he will outlive all of his friends. This could cause him to be more withdrawn at first. But then, as he interacts with the students at Westover Hall, he might find new meaning in his immortality. He could start to see it as a way to protect others not just for a short time, but for as long as he exists. He could also form deep and lasting friendships with those who accept him for who he is, regardless of his immortality.
In the fanfiction, Percy might face new challenges related to his demigod status. Maybe he has to deal with the aftermath of the war, like rebuilding Camp Half - Blood or helping new demigods adjust.
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.