Should I ask my cousin to be friends on facebook? His parents dont talk to my family anymore?

You should definitely! He doesn't have to add you if he doesn't want to. Also, was it that the uncle that stopped talking and had issues? Then the son should be able to make his own decision. Maybe you should mention it to your mom. She might not have problems with it especially if she's upset that he stopped talking. I think maybe you can try when neither of you are still living with and depending on your parents. Good luck!