It is permissible for your father to put the house asked about in your mother's name and to make that a pure gift or to do it in return for what she has of the mahr and the delayed portion, if applicable.
However some things must be borne in mind:
That your mother is no longer regarded as his wife, so if your father has another wife, he is not obliged to treat them equally, because she is no longer his wife. If he does not have another wife, then there is nothing to prevent him from giving a house or anything else to his ex-wife.
If he wants to put the house in your name, that is in the name of his children, if he does not have any children from another wife, then there is nothing wrong with him putting it in your name, because a man may give his children whatever he wants. If he does have other children, then he has to treat them fairly in terms of giving, so if he gives something to one of them, he has to give the others something similar. See the answer to questions no. 22169 and 89720
If your father does not own any part of the house, either the land or the structure, and it was built for you by well-wishers, then it is your property and your father has nothing to do with it.
If the land belonged to your father, then the same applies as stated above: it is permissible for him to register it in his ex-wife's name or in his children's names, if he does not have other children, or if he has other children and he treats all the children fairly in terms of giving.
We should point out that it is not permissible to accuse your father of zina and that this is regarded as slander, so long as you have not produced four witnesses to that, because Allah, may He be exalted, says (interpretation of the meaning):
And those who accuse chaste women, and produce not four witnesses, flog them with eighty stripes, and reject their testimony forever, they indeed are the Fasiqoon (liars, rebellious, disobedient to Allah).
Except those who repent thereafter and do righteous deeds, (for such) verily, Allah is Oft-Forgiving, Most Merciful.
Moreover, the father has the right to be treated with kindness and respect, and to have ties upheld with him, even if he is an evildoer or wrongdoer.
We ask Allah to set straight the affairs of all the Muslims.
And Allah knows best.
Reproduced from Islam QA