Does a mother's desire to write over a portion of her wealth to her son who cares for her constitute a bequest to an heir, and would she be sinful if she did so, knowing that he refuses this and believes that Allah will compensate him?

If the mother, by writing, intended to earmark something from her possessions for her son after her death, then this is a will in favor of an heir, which is not valid unless all heirs approve it. However, if she intended a gift during her lifetime, the preponderant opinion is that it is not permissible to single out some children for a gift, even if one is dutiful. Serving and caring for parents is obligatory upon all children, and if one of the sons and his wife undertake this alone, seeking Allah's pleasure, then they will have a great reward.