Is the money that the cousin won from the lottery halal or haram, knowing that he was unaware of his participation in it and that it came through immigration papers?

Dealing in lotteries is impermissible because it is gambling. The fact that your cousin was unaware of this absolves him of sin, but it does not make it permissible for him to benefit from theem, because it is an ill-gotten gain. This money must be spent on charitable causes for the poor and needy, for ridding oneself of unlawful wealth is achieved by spending it on acts of charity.