What is the ruling concerning this gift presented to memorizers of the Noble Qur’an, which the father received on behalf of his daughter who has memorized half of the Qur’an, knowing that she has not completed its memorization?

If the donor specifies that the gifts are for those who have memorized the Book of Allah (the Qur'an), then it is not permissible for the Sheikh to give them to someone who has not memorized it, nor is it permissible for the non-memorizer to take them. This is because the condition set by the donor is legally binding, and the gifts must be given only to the memorizers. However, if the donor intends for the gifts to be given to those who have memorized a portion of the Qur'an, or if they delegate the matter to the Sheikh, then it is permissible to take them.