Is it permissible for a Muslim to write a will for his son asking him to pray for his forgiveness and mercy on a specific day each year after his his death?

There is no harm in children enjoining mercy upon their parents and supplicating for them, because it is in accordance with the divine command and with what is authentically reported from the Prophet, peace and blessings be upon him, that "a righteous child who supplicates for him" benefits the deceased. And that among the acts of filial piety towards parents after their death are: "praying over them and seeking forgiveness for them." Designating a specific day for supplication is not harmful if it is one of the virtuous days; otherwise, we do not recommend designating a specific day, and it suffices to enjoin supplication without specifying a date.