Has he attempted at all to find work? Is it possible his mental health issues interfere with getting a job? If so having him apply for SSI might be an option, it is possible to get it for mental health issues if they prevent you from holding a job. Sometimes throwing someone out isn't the best option, especially if they don't have any resources or places to go...i mean otherwise that being homeless with nowhere to go which not everyone is going to be able to pull themself out of but I am sure you are aware of that. But it is clear there needs to be some kind of change.

I'm 24 and live at home with my mom, and if she threw me out I'd probably end up pretty bad off...but I am working on getting on SSI and going to therapy so she can see that I am doing something.

Its understandable your frusterated, and I wish I knew of a solution but I don't know your full situation or how exactly the mental issues interfere with your sons functioning.