Will she be sinning if she refuses to give up her right? Will she be disobeying her mother if she does not respond to her demand?
The basic principle is that whatever is in a person’s possession when he is alive belongs to his heirs after he dies. Allah, may He be exalted, says (interpretation of the meaning):
“There is a share for men and a share for women from what is left by parents and those nearest related, whether, the property be small or large - a legal share”
Hence it is not permissible for anyone to take the share of one of the heirs or to demand that he should give up his share of the inheritance unless he agrees willingly to do that, because that comes under the heading of wrongdoing and transgression, and the Prophet (blessings and peace of Allah be upon him) said: “It is not permissible to take the wealth of a person except with his willing consent.” Narrated by Ahmad, 20172; classed as saheeh by al-Albaani in Saheeh al-Jaami‘, no. 2780
With regard to your wife’s mother’s claim that she owns that part of your wife’s share, if she does not have any proof of the validity of her claim, then your wife does not have to give up her rightful share of the inheritance. Not giving up that share is not regarded as disobedience to parents at all.
Saying that the mother is not entitled to that part of the inheritance does not imply that she should not honour her mother and treat her kindly; rather your wife should speak kindly to her mother and ask good and righteous people to intervene in order to solve this problem, in the hope that Allah may guide the mother to truth and right guidance.
If your wife willingly gives up that part (of the inheritance), or some of it, to her mother, seeking reward from Allah, then this is good and an act of kindness, in sha Allah. But, as stated above, she is not obliged to do so and she is not sinning if she does not agree to her mother’s request.
And Allah knows best.