I signed up with Adsense a long time ago(about 5-6 years) and got banned. I'm not sure what for but, it's not letting me reapply because of this. My question is: if I were to sign up using a family member's name, would I need to use their Social Security also?
for sure, you will have to provide the account holder's SSN otherwise its a tax fraud and adsense will keep track of your unique SSN
Absolutely they keep track of SSN. I would e-mail them first before signing up under a relatives name. You might be able to get in again if you explain things have changed.