Answer
At the outset we wish to advise that it is not permissible for a Muslim Female to marry a Non-muslim male. Your cousin should continue advising the man she is staying with to accept Islam. If she is an upright Muslimah who constantly advises him of Islam and makes Du’aa to Allah, then, Inshallah he will accept Islam. Until then it is not permissible for her to live with him.
If your cousin passes away with the believe that there is no diety except Allah and Muhammad (Sallallaahu Alayhi Wasallam) is the final messenger, she will pass away as a muslim inshallah.
and Allah Ta’ala Knows Best
Mufti Ebrahim Desai