What is the Shar’i (religiously lawful) solution for the problem of a man who divorced his wife, whose family had lent him a sum of money to buy a house, and he is now unable to return the amount in cash, and wishes to register a portion of the house in her name equivalent to the amount, but she refuses and wants the amount in cash?

You must repay the loan in kind. You are not obligated to compel your wife's family to accept a share of the house in exchange for the loan unless both parties agree, as it is a sale that is not permissible without mutual consent.

If you are in financial hardship and do not possess what can cover the debt in excess of your basic needs, then you must be given a reprieve until you are able to repay. However, if you have assets that can be sold to repay the debt, then you are not considered in hardship. The criterion for financial hardship is that the debtor does not possess wealth beyond their basic needs that would suffice to cover their debt.