If the person wished to join a lineage society through the father's ancestors (say they found a revolutionary war soldier or a mayflower ancestor), proof of parentage would be required. There would be only two ways to do this. 1. File a correction to the birth certificate and the mother would have to have a sworn statement to do that. 2. File a sworn statement from the mother with the lineage papers.
Either way, that statement would be needed. If the mother is willing, it would be good to go ahead and correct that birth certificate. Once she dies, there is probably no way to do so.