A cousin (even a first cousin) is not considered "closely related" for tax purposes. Hence, the additional requirement that he live with you all year to qualify as your dependent.
If he does not qualify as your dependent, he (or whomever does claim him as a dependent) may claim* the tuition credit based on the tuition you paid.
Although the general rule, in taxes, is that you must be the one making the payment, to get the deduction or credit, there is an exception for education.
*One caveat: A full time unmarried student, under age 24, even if he doesn't qualify as a dependent, is only eligible for the refundable portion of the American Opportunity Credit if he supports himself by working. You cannot be supporting yourself on support from relatives, 529 plans or student loans & grants. It is usually best if the parent claims that credit.
You cannot claim a credit if you are, or can be, claimed as a dependent by someone else.
Reference: Line 7 instructions for form 8863. https://www.irs.gov/instructions/i8863
