Is it obligatory for a mother to fast on Friday morning so that her deceased son may find food?

What you mentioned about your mother not eating on Friday morning has no basis in religion; rather, it is a superstition that a Muslim must abandon. What she should do is supplicate for her son and give charity on his behalf to the poor and needy. Charity given on behalf of the deceased benefits them, and its reward reaches them, as mentioned in the hadith: "My father died and left wealth, but he did not make a will. Will it expiate for him if I give charity on his behalf? He said: 'Yes.'" And his saying: "My mother died suddenly, and I think that if she had spoken, she would have given charity. So, will I have a reward if I give charity on her behalf? He said: 'Yes.'"