My mother wants to know if she can give me (her daughter) the house where we live as a Hiba (gift)? I am one of two daughters; my sister married last year and doesn't have a good relationship with our mother. I am studying and my mother wants me to have some security for my future.

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

It is not permissible for your mother to give her house to you and not give anything to your sister who is her other daughter.
Jaabir narrated that the wife of Basheer said (to her husband): "Give to my son your slave as a gift, and make for me Allah's Messenger () a witness. He came to Allah's Messenger () and said: “The daughter of so and so (his wife ‘Amrah bint Rawaahah) asked me to give my slave as a gift to her son, and call for her Allah's Messenger () as a witness. Thereupon, the Prophet said: “Does he (An-Nu'maan) have brothers? He (Basheer) said: ‘Yes.’ He (the Prophet) said: ‘Have you given to all others (his brothers) as you have given to him? He said: ‘No.’ Thereupon, the Prophet said: “Then it is not fair; and verily, I can bear witness only to what is just.” [Al-Bukhari and Muslim]

Therefore, you are demanded to try to reconcile the relations between your mother and your sister till they become as smooth as they should be. Also bring to the knowledge of your mother that not having good relations with her daughter does not justify that she be unfair with her as far as gifts are concerned. If she (the mother) deals with her daughter unjustly this may cause the daughter to be undutiful to her mother. Thus, the mother cooperates with her in that bad practice. Allah Says (what means): {… but do not help one another in sin and transgression.} [Quran 5:2]

Allah knows best.