What is a word for "the desire to be considered a victim"?
The expression self pity is perhaps more accurate than "masochism".
Masochism has certain additional connotations, quite separate from the desire to garner sympathy or discharge perceived guilt.
Some friends have suggested "martyr complex" and "victim complex". Those are pretty good, let's see if any other suggestions come up.
While technically the act rather than the desire, "victim playing" seems to be a good fit.