I am probably the wrong person to reply to this question as self pity is not something that I instinctively have much time for or sympathy with.
It may of course be the result of a depressive condition in which case a little tact and diplomacy may be warranted or even a gentle push towards seeking some professional help for the party concerned.
Other than that, I see self-pity as an unpleasant and manipulative trait used by basically selfish individuals to try to get their own way in any given situation. That may sound a bit harsh but to concentrate solely on yourself, and generally to the exclusion of others can only be described as selfish.
I would also go as far as to say that self pity is a psychopathic personality trait, inasmuch as these people rarely show any empathy or care for the problems of others within their social circle. As for what to do with them, I think I would just point out how many things they have to be thankful for and suggest they stop moaning.