In case that reality is exactly as that which is mentioned in the question, the last living entrusted person has to
execute the old lady's will and perform Hajj on her behalf before he dies, just like all those who were mentioned in the question. However, he should do so intending Hajj on behalf of his concerned great grandmother though he does not know her name. Thus he should say when entering the state of Ihram (ritual state for Hajj and `Umrah) from the Miqat (sites for entering the ritual state for Hajj and `Umrah): "I intend to perform Hajj on behalf of the first entrusting person which is my great grandmother; the mother of my paternal grand mother." On the other hand, Sadaqah is not a sufficient substitute for performing Hajj on her behalf. May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.