Must we approve of our 54-year-old mother marrying a man whom her children do not approve of, especially since the youngest brother does not accept him, or should we demand that she wait, out of consideration for the brother's feelings and to appease him and the mother's family?

If your mother wishes to marry, you do not have the right to object or delay her marriage to appease your brother. Rather, you can advise her to choose a righteous husband. If the man is competent (and competence here refers to religious competence), then she should not be prevented from marrying, otherwise, that would be an act of 'adhl (unjust prevention of marriage) that would permit her to take her case to a judge. Remember that the mother's right is great, and honoring her is among the best acts of drawing near to Allah. Your objection to her marriage on the pretext of her age is unfounded, especially if she is in need of a husband.