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.