Really I think that if you're going to make a character with mental illness, you need to follow two simple rules: 1) try to be accurate if you're going to label it with a specific term, and 2) don't be an attention whore.

One of my more notable characters that I've had for a long time, tweaked and recycled a couple times is a sociopath. He's incapable of empathy or caring for other individuals. And... that's really all that I've given him in terms of that attribute. I don't think I've ever even technically referred to him as a "sociopath" in the RP, now that I think about it. He just comes off to others as really cold, sadistic and somewhat of a troll - not that he's like that because he's a sociopath, though, those are just other parts of his personality. He also has... I dunno, I guess you would call it a "God complex"? Basically he considers himself a god and is always trying to assume complete control of everything around him - which makes sense considering that, in most incarnations of his character, he usually has an admittedly ridiculous powerset and a whole world he's created, thus putting him only a stone's throw away from being a god. Though in other incarnations where he's less powerful, I guess it falls under the category of delusional. Also, in his latest incarnation, I've given him some early signs of alzheimer's. Every once in a while he has these episodes of memory failure, but, again, I don't think I've ever used the term in the RP. And to the other RPers, he probably just looks like a forgetful person. 

Another notable example is a character I made for an RP that fell apart almost immediately after it started, so I never got much of a chance to play her, but I had a lot of fun designing her. Basically the RP had a sort of virtual reality-type setting, and my character was basically supposed to be a glitch in the system. As a result, she had a lot of "mental problems", but I never really labeled any of her quirks with the names of actual mental disorders. Some of them were inspired by real mental disorders, and a few other qualities of hers were just things that I thought would be fun to play. Her awkward social interactions were inspired by autism/aspergers, her emotional instability was inspired by bipolar disorder, etc, but I knew that I wasn't portraying any of these things perfectly accurately - in fact in some cases I intentionally let her traits differ from what I knew the actual disorder(s) looked like. But that was ok, because nothing she had was labeled with the name of a real disorder. They were all just generically lumped under the term "mental disorders", which I explained through the fact that her virtual brain just wasn't wired the right way. And I think that's a good way to get around portraying mental disorders inaccurately, intentionally or not - just don't label it as such. With RP's, you have lots of reasons to justify portraying a mental disorder that doesn't actually exist. And so long as you don't name it as something that you were inspired by or that you think it [i]sort of[/i] looks like, then it shouldn't be offensive to anyone. Or, at least, that's my take on it.

Also, I agree that characters with mental disorders can be really fun to play. They let you experiment with a lot of social interactions and characters' thoughts that you wouldn't otherwise get the chance to play through.