If you are ready to pay, you can open a joint account. This means that there will be one credit limit, one balance, one payment, and one due date for you and your cousin. You both will be making charges on the credit card. However, you will be paying the bill. You both will establish credit history as it will show up on both credit reports.
Another option is to add your cousin as an authorized user. But he will have no legal obligation to pay back the credit card debt he charged because only the cardholder is legally responsible.
Disclaimer: This editorial content is not provided or commissioned by the credit card issuer(s). Opinions expressed here are the author's alone, not those of the credit card issuer(s), and have not been reviewed, approved or otherwise endorsed by the credit card issuer(s). Reasonable efforts are made to present accurate information, however all information is presented without warranty. Consult a card's issuing bank for the terms & conditions.
All rates and fees, and other terms and conditions of the products mentioned in this article/post are actual as of the last update date but are subject to change. See the current products' Terms & Conditions on the issuing banks' websites.