Praise be to Allah.
If the apartment referred to in the question belonged to your mother, then the apartment must be shared out among her heirs according to the Islamic guidelines on division of inheritance, and each person who is entitled should be given his rightful share. In this case, your sister’s request for her share of the house – and not just of the rent – is a legitimate request.
But if the apartment belongs to your father, then she has no right to demand any share of the rent of the apartment, because it is in the possession of your father, and he may dispose of the rent just as he may dispose of any other wealth he has, and she has no rights over her father except maintenance, if she is not married, in which case he should spend on her as he spends on his other children, each according to his or her need. If your father wants to give her a gift of money or property, then he must treat her and all her brothers and sisters fairly. As for the fact that you live in his house, that does not give her any right to the rent of the lower apartment, and it does not give her any right to your father’s wealth in general, except obligatory spending on her maintenance, as referred to above.
During a person’s lifetime his wealth cannot be described as an inheritance, and those who will be given a share of it cannot be called heirs, because the first condition of inheritance is the death of the testator. There are two issues that are worth pointing out here:
1.If your father wants to give his children – only – something whilst he is still alive, that is described as a gift or present, and it is permissible for the father to do that, but that is on condition that he treat all his children fairly, both male and female.
2.If your father wants to divide his property among all his heirs, it is permissible for him to do that on condition that he does not deprive some of them and he does not detract from their rights. However we do not encourage anyone to do that, because of what it may cause of some of the children being tempted by that wealth into falling short in honouring their father. Moreover, he himself may need that wealth after it is no longer his. He may have more children, other than those to whom he gave that wealth, in which case those new children would be deprived of having a share of their father’s wealth with their siblings.
It should be noted that if handover of a gift to children is deferred until after one dies, then it becomes a bequest, and it is proven in the hadith that “There is no bequest to an heir.” Such a gift should be put back with the estate and shared out according to the laws prescribed in Islam.
For more information, please see the answer to question no. 71297
And Allah knows best.