Most Helpful Girl
Some might, but I think it also depends on numerous factors such as what the mental health issues are and if he is on medication to control/help those issues. Some mental issues are way more severe than others. Overall though, I personally don't think I would be interested in dating someone with a mental illness.
Most Helpful Guy
They can be, yes. I have social anxiety and depression and have a couple of girls that am about to start dating. Only one of them knows it and is fine with it, supportive of my improvements, the fact I'm getting help etc. But, I wouldn't go advertising your problems, that will send them running. Get to know them a bit first. If they see positive qualities they like in you, they may be able to look past your challenges. If they don't notice it, its not so severe that you can't date. If they do, you explain it to them, and what you're doing to improve.