Is it permissible for my mother to perform the missed prayers on behalf of my deceased grandfather, and what is permissible to dedicate its reward to the deceased?

It is not prescribed to make up prayers on behalf of the deceased, and neither you nor your mother nor anyone else may make up the prayers of your deceased grandfather. It is permissible to dedicate the reward of any act of worship to the deceased, and the deceased benefits from supplication, seeking forgiveness, and charity given on their behalf, by scholarly consensus. It is not advisable to intend to dedicate the reward of every act of worship to the deceased, for fear of depriving the one performing the act of its reward. It was not the practice of the righteous predecessors to share the reward of supererogatory acts of worship with others. It is best to confine oneself to what the righteous predecessors followed, for all good lies in following, and all evil in innovation. Acts of worship are based on divine textual evidence.