What is the duty of a granddaughter whose grandfather bequeathed all his property to her without the knowledge of her disobedient mother, and is it permissible for her to take this money?

If the grandfather wishes for the ownership of his properties to transfer to his granddaughter after his death, this is considered a will (wasiyyah). A will is only enforceable up to one-third of the estate. Anything exceeding one-third belongs to the heirs. If there are no heirs other than the daughter, then the remaining wealth is hers.