If his mom wants him to start dating someone that he is related to by blood, his mom might want to consider seeing a therapist. In some parts of the country, that's considered incest.
If he doesn't want to date his cousin, he should say so. Unless he's under the age of 18, he isn't obligated to do anything his parents tell him to do...and even if he is under 18, there are limits to what a parent can MAKE him do.
If you don't feel that the relationship is going well, re-evaluate it and decide if it's worth staying in.
How long have you been dating? How long has the relationship been long-distance? How often do you talk to or see each other? What type of closeness did you have before the relationship became long-distance? Have YOU cheated on him (if you have at all, even once, then you aren't as committed as you think)?
Long-distance relationships almost never work because the people involved lack emotional maturity, for the most part, and also have a constant struggle with trust and honesty. If you are having doubts, get out of it and find someone you can have a REAL relationship with.