For all the sentimental reasons (and just because I love it) What's Eating Gilbert Grape?
If I had to pick favourites I'd go with Fight Club and American Psycho. I love both films so much and have seen them countless times and each time I watch them I take away something new and thought provoking. Not to mention both of them include incredible performances from the cast - Brad Pitt (as Tyler Durden, one of the greatest movie characters of all time) + Ed Norton, and then Christian Bale as Patrick Bateman - that really make them something special. Honourable mentions go to Awakenings, Donnie Darko, and Eternal Sunshine of the Spotless Mind.
I'm surprised nobody has mentioned Rain Man.
First off, all great choices man, all great films. Donnie Darko is my second favorite film ever made (MAYBE third...not sure, it's in there somewhere with my other top two films, Blue Velvet and Halloween), and Eternal Sunshine of the Spotless Mind rounds out my top ten as well. I'm a HUGE Charlie Kaufman fan (writer of that film), and few films have ever left such an emotional impact on me as Sunshine of the Spotless Mind did. But, I mean technically it's not really a mental illness movie. All of the hallucinations and the dreamlike state the film takes place in is all because of the treatment being given to Carrey and Winslet's characters, technically they aren't really mentally ill. I mean I suppose you could argue that Clementine was mentally unstable and Carrey was anti-social but I'm not sure I'd qualify those under this category that TM is looking for. Amazing film regardless.
A few mentions from myself here; drug addiction being in large part a mental disorder, I would consider Requiem For a Dream for this category. This is by far the greatest film ever made about drug addiction, take it from someone familiar with the subject, no film has ever captured the highs, the lows, and the sheer fucking horror as well as this film. I'd go so far as to call it one of the most frightening films ever made, and it's not even remotely a horror film!
Speaking of Charlie Kaufman from earlier, Adaptation is another film worth mentioning here. One of Cage's very best performances (Yes, he can in fact act his fucking ass off, if you don't believe it watch Adaptation or Leaving Las Vegas) and a film that captures that famous Kaufman bittersweet romanticism and the surreal settings that can come with mental illnesses and disorders. Another amazing film.
As Cookie mentioned, Awakenings is another great film. I absolutely despise Penny Marshall but damnit if she didn't make a wonderful film there, although that's mostly because of the amazing performances by both Robert De Niro and Robin Williams. I really, really have a special place in my heart for that film.
I'm sure there are plenty of others I could name, and I know I'm forgetting them. One I'd like to mention is the film Maniac, which is a low budget slasher film made in the early 80s which in large part delivered one of the most realistic and somehow sympathetic portrayals of a serial killer/maniac. The performance by Joe Spinell is astounding.
Speaking of serial killers films, I wouldn't be able to forgive myself if I didn't mention Henry: Portrait of a Serial Killer, which take that same formula that I just described with Maniac, cuts out all of the cheesy and fat parts, and creates one of the most disturbing and visceral films ever made. Michael Rooker's performance fucking SCREAMS Oscar and it's a god damn shame he wasn't recognized for his role here as Henry Lee Lucas, the real life serial killer who claimed to have killed hundreds of people.
Another film, which is a much more recent one, worth mentioning is William Friedkin's film Bug, starring Ashley Judd and the absolutely amazing Michael Shannon. That film really just got under my skin (no pun intended) like few I've seen over the last few years, and the film really is just like a fucking punch in the stomach. It's brilliant stuff.
Right, I'll stop there, because now I'm just naming a shitload of films. Rest assured if I get bored enough I'll come back and bring up some more.