What is the fate of the money saved by the grandmother from the rent of her son's house after her death: Is it for all the heirs, or only for her son who used to give her the house rent, whether the grandmother stated that or not?

The ruling on the money your father gives to his mother differs according to his intention:

If he intended it as a gift, then it becomes the property of your grandmother, and it is included in her estate upon her death and is divided among the heirs according to Islamic law. However, if she bequeathed that the money, after her death, be for your father, then this is a bequest to an heir which is not binding on the other heirs except with their consent.

But if his intention was merely for her to benefit from it without the intention of gifting, then whatever remains of it remains the property of your father and is not included in your grandmother's estate.