Is it permissible, Islamically and legally, for the grandmother to relinquish her house to her daughter, who is not a legal heir, while there is another legal heir, her son?

The legal ruling regarding the house depends on its ownership. If it was owned by the grandfather, and his heirs are limited to his wife and son, then the wife receives one-eighth, and the remainder goes to the son. The wife is only entitled to dispose of her one-eighth share. The mother must be fair among her children in giving gifts, unless allocating a specific gift to a daughter is due to a need or poverty. A gift is not effective unless it is completed and possessed by the recipient during the lifetime of the giver. Disputes concerning inheritances should be referred to a judge.