What is the ruling on paying money to a friend to file a lawsuit to collect one's inheritance, in exchange for a percentage of this inheritance?

It is not permissible to agree on an increase over the loaned money, as that is usury (riba). It is permissible to lend money only without an increase, or to enter into a financing transaction such as Tawarruq, where you sell a commodity that the needy person buys to resell it and benefit from its price. It is also permissible for the debtor to sell you a described commodity that is under his liability for a specific price that you pay him now, then he buys the commodity for you when its due date arrives.