Should I ask my cousin to be friends on facebook? (It's more complicated then you think..)?

Send a friend request and try. The worst that can happen is that he will ignore it. It's yours or your cousin's faults that the family isn't talking. You don't need to ask your mom either; there's no reason for you not to be in contact with your cousin. A similar thing happened in my family and we weren't on speaking terms with my cousins for over 5 years. And you know what, now that we do speak again, I like having my cousins around.