Is prayer obligatory for a grandfather who has begun to lose his mind, imagines things, and has no sense of time? And is it permissible for the mother not to respond to his repeated calls if she is certain that he has no urgent need, especially in the middle of the night?

Prayer is not obligatory for your grandfather due to his mental impairment, as the Prophet (peace and blessings be upon him) said: "The pen has been lifted from three: from the sleeper until he awakens, from the child until he reaches puberty, and from the insane person until he regains his sanity." A narration also mentions "and the senile person," which is sound in meaning even if its chain of transmission is weak. Senility is the deterioration of the mind due to old age. Therefore, whoever reaches a stage of senility where he no longer perceives time or distinguishes between prayers is not obligated to pray. If your mother attends to your grandfather's needs and it is highly probable that his calling out is not for a necessity, then there is no blame upon her for not answering him if it causes her hardship.