What is the ruling on a father singling out a prize for a child of his who has memorized the Qur’an, and is the money forbidden or its utilization unlawful if the father does not act justly in giving to his children?

A father's designation of a reward for his children's memorization of the Qur'an is a ju'l (a stipulated payment for a specific act), not a pure gift. Even if the father does not specify a reward but gives his son a gift for memorizing the Qur'an, there is no harm, because favoritism in giving gifts is permissible for a legitimate Sharia reason, such as engagement in knowledge or memorization of the Qur'an. If the favoritism is without a legitimate reason, the gift is valid according to the majority of jurists, and it is merely disliked (makruh), and neither the parent nor the child incurs sin for it.