Whenever I answer a question like this, I'm deeply aware that someone may be reading this who is seriously contemplating suicide—and I would do everything in my power to convince them not to take that final, drastic step. With Christ there is always hope.
God loves them; He loves them so much that Jesus Christ was willing to die on the cross for their salvation. And others care about them, as well (even if they have a hard time believing it). Suicide is never God's will, and even in the midst of life's worst storms we can look to Christ and discover that He truly is "an anchor for the soul, firm and secure" (Hebrews 6:19). The first step back from the brink of suicide is to turn to Christ and open our hearts to Him.
Suicide is always a tragedy—but in itself it is not the unpardonable sin. The only sin God cannot pardon is the sin of rejecting Him. God knew your cousin's heart, and He also knows that mental illness sometimes clouds a person's judgment so much that they aren't fully responsible for their actions.
At the same time, ask God to help you be sensitive to the needs of others—especially someone who may be facing discouragement or depression. Often simply knowing that someone cares will help them turn the corner.