There, you said it. That is the problem. Your cousin, aka her husband, gave his wife your contact info and told her to become your friend. Perhaps she is doing just that and trying to be an obedient wife. I don't think she is the annoying one. I think its your cousin who gave your personal info to her without consulting you first. You said he was "thrilled" after she contacted you. Don't you think he told his wife that he gets happy when she communicates with you? Its not her fault. She is just trying to make her hubby happy.
You know when I first read this, this exact thought was in my head. That is a possibility, that it is not your cousin in law's fault, especially because she did not ask you for the contact information.
Regardless, if you don't like replying to her e-mails or facebook message you can always give her the "busy-busy" signal. But if you do that, just occasionally call her up, to see how's she's doing, just so she knows you and her are in good terms. I think this sitation is a temporary problem- after a while this will all pass when she gets going with her own life and starts to feel comfortable with her new family- so be as nice as possible.