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.