What are my options if I'm being sued for $1200 for a credit card obtained without my knowledge by a family member?

You need to decide whether you want to allege ID theft. Normally they will want you to agree to prosecute the culprit. But that will get the case dismissed. If you are being sued by the original creditor and do not claim ID theft, they may be able to prove a case. If it is a debt buyer, they may not. You should consult an attorney who handles such cases.