Is it permissible for a son to complete the procedures for withdrawing a gift that his father had given to his mother, which the father was unable to finalize before his death, especially given that the mother abandoned the father years after the gift was made?

It is permissible to retract a gift before its possession, because it does not become binding until it is possessed. If possession has occurred, the general rule is that it is forbidden to retract the gift, with two exceptions: a parent retracting what they gifted to their child, based on the hadith: "It is not permissible for a man to give a gift or bestow a present and then retract it, except for a parent in what he gives to his child..." And if someone gives a gift for a specific purpose and that purpose is not achieved, they may retract it, such as a gift given for a reward; if the reward is not obtained, they may retract it. If the donor dies before retracting the gift, then if they had demanded the gift back or filed a lawsuit to reclaim it, the heirs may pursue the claim; otherwise, they may not.