Is it permissible to take the life insurance payment from the university where the father used to work, given that the university insured his life and not he himself, and that the family is in dire need of these funds?

If the insurance that the university subscribed to for your father was a commercial, forbidden insurance, then the insurance amount exceeding the paid premium is considered unlawful wealth. It must be disposed of by spending it on the interests of Muslims or giving it to the poor and needy. Since you are poor and in need, it is permissible for you to take what suffices your need from it, as you meet the description of eligibility, as mentioned by An-Nawawi. However, if the insurance was cooperative takaful, then there is no harm in benefiting from it.