Don't let his arguments shake your faith or make you wonder if he might be right after all for he isn't. Remember: The Bible tells us that "The fool says in his heart, 'There is no God' " (Psalm 14:1).
You see, the problem for most atheists isn't a lack of evidence for God's existence. In fact, the evidence is so overwhelming that anyone who honestly looks at it with an open mind will conclude that God must exist. As the Bible says, "Since the creation of the world God's invisible qualities his eternal power and divine nature have been clearly seen, being understood from what has been made, so that men are without excuse" (Romans 1:20).
What, then, is the problem for most atheists? In my experience the problem is their will that is, they don't want to believe in God. They know that if they admit God exists, then they can't go on living any way they want to but must humble themselves before Him. And that is something they refuse to do.
Pray for your cousin; only God can change his heart and show him his need of God. In addition, submit your own life more fully to Christ, and ask Him to help you be an example to your cousin. No one can argue with a changed life! Also, ask your local Christian bookstore to recommend books that answer the questions he is raising.