Praise be to Allah.
If the husband has died and the woman wants to marry someone who is religiously committed and of good character, the guardian is obliged to arrange that marriage, whether the guardian is her father or her son, if the father is not there. And it is not permissible for anyone to prevent her, because this is one of her rights. It makes no difference whether the woman is old or young, or whether she has children or not.
The fact that the woman is old is not a barrier to her getting married, because there are many benefits in marriage, such as the woman’s need for someone to look after her affairs, take care of her, and keep her company in her house, instead of her living alone in her house with no company.
What you have to do is ask about this man who has proposed to your mother; if he is good in terms of religious commitment and character, then you have to arrange the marriage and it is not permissible for you to refuse to do so.
And Allah knows best.