For the reservation to avoid repricing, at least one person on the original reservation must sail. So if everyone originally booked on that reservation are now NOT going, you cannot substitute names and keep the same pricing.
What would happen if you keep your mother's name on the booking and then she is a no-show ? Truly, I don't know. Maybe they would not notice. Maybe they would. What they would do about it if they DO notice....is a guess. I suppose you could take that chance and see. ....but you should be prepared to have to reprice the booking on the spot. Not sure they would go to that trouble, but who knows,,,