What is the ruling on the pension of a retired mother who used to work as an accountant in a usurious bank, and is this money considered unlawful, preventing a pious young man from marrying her daughter or eating in her parents' home?

Working in a usurious bank is forbidden (haram), and its salary is forbidden, as is the retirement pension, because it is a consequence of a forbidden contract. Your friend's mother must donate what she takes to the poor and needy and for the general welfare of Muslims. She should not benefit from it unless she is poor, in which case she may take from it according to her needs. There is no objection to marrying her daughter. As for eating at her place, if her money is mixed (lawful and unlawful), there is no objection to it. But if all her money is unlawful, then eating from it is forbidden. If most of it is unlawful, then it is disliked (makrooh).