Is it permissible for a mother to gift half of a house to her son? And would this be considered an injustice to his siblings, given that the house is registered in the mother's name? And is the son obligated to demand his share to avoid disputes, especially since the mother promised him that?

The fundamental principle is the obligation to be just among children in giving, and it is not permissible to favor one of them with a gift unless there is a legitimate reason. Among the legitimate reasons are:

1. The need of one of the children, such as illness, a large family, or engagement in seeking knowledge. 2. The transgression or innovation of some children, or their use of the gift in disobedience. 3. If a son gives gifts to his father and helps him with building expenses and the like, it is permissible to allocate to him an amount commensurate with his assistance, by way of recompense.

And if the rest of the children agree to one of them being singled out for a gift, there is no harm in that.