In general, a contract is personal to the person signing it. There must be mutual assent between the two sides making the agreement. Without this element, there is no contractual relationship. Unless the family was a participating party to the contract, they are not responsible for the obligation. Having said that, the person who died would have an estate that has to be settled, and if he or she were contractually bound to a lease, you would need to review the contract carefully for language addressing what happens if the person dies. Depending on the circumstances and the jurisdiction, the contract could die with the signor, or if the lease becomes a debt against the estate, then the debt would be paid from proceeds from the administration of the decedent's estate. There are specific statutes related to creditor's claims against a decedent's estate, that have to be strictly followed, and if the creditor fails to comply with these rules, again depending on the state, the creditor's claim could be barred. There may also be rules in your state that apply strict liability to the estate and the personal representative of the estate. You would need to reveiw such rules with a probate and estate attorney in your state.