Praise be to Allah.
Firstly: The basic principle is that if a woman is divorced after the marriage has been consummated, she is entitled to keep the mahr in full. If the husband did not divorce her until after the mahr was returned to him, this is khula’, which is permissible in return for the mahr, or more or less than that.
Based on this, your relative does not have to return the ten thousand to her ex-husband, so long as they agreed that she can keep it.
Secondly: If the woman attributed words or deeds to her husband of which he was innocent, then this is a serious lie and is haraam. What she has to do is to repent to Allaah and pray for her husband and for forgiveness for him. She does not have to tell him about that or ask for his forgiveness. The same applies to anyone who backbites or tells lies about another person; it is not a condition of repentance that he should tell him about that, because of the enmity and hatred that that may lead to.
Ibn al-Qayyim (may Allaah have mercy on him) said in al-Waabil al-Sayyib, p. 219:
Chapter 56: What a person who has gossiped about his Muslim brother should say:
It is narrated from the Prophet (peace and blessings of Allaah be upon him) that the expiation for backbiting is to pray for forgiveness for the one concerning whom you gossiped. You should say: O Allaah, forgive us and him. This was narrated by al-Bayhaqi in al-Da’waat al-Kabeer, but he said there is some weakness in its isnaad.
There are two scholarly opinions concerning this issue, both of which were narrated from Imam Ahmad. The issue is: in repenting from backbiting is it sufficient to pray for forgiveness for the one about whom you gossiped, or must you also tell him and ask for his forgiveness?
The correct view is that there is no need to tell him, rather it is sufficient to pray for forgiveness and to speak of his good points in the places where you gossiped about him. This is the view favoured by Shaykh al-Islam Ibn Taymiyah and others. Those who say that it is essential to tell him regarded backbiting as being like financial rights, but the difference between them is obvious. In the case of financial rights, the person who is wronged benefits from the restoration of his rights; if he wishes he may keep it and if he wishes he may give it in charity. But in the case of backbiting, that is not possible and the only thing that results from telling him is the opposite of what the Lawgiver seeks, for it will upset him and hurt his feelings if he hears what was said about him, and it may stir up enmity that can never be assuaged, and there is no point in that. So the wise Lawgiver would never permit or allow it, let alone enjoin it or command it. The aim of sharee’ah is to reduce and eliminate evil, not stir it up and make it worse. And Allaah knows best.
Al-Safaareeni said in Sharh Manzoomah al-Adaab, 2/577.
Ibn ‘Abd al-Barr says in his book Bahjat al-Majaalis: Hudhayfah (may Allaah be pleased with him) said: The expiation of one who has gossiped is to pray for forgiveness (for that person). ‘Abd-Allaah ibn al-Mubaarak said: The way to repent from backbiting is to pray for forgiveness for the one about whom you gossiped.
It says in al-Adaab al-Kubra something similar to what Ibn al-Mubaarak said, and this was the view favoured by Shaykh Taqiy al-Deen and by Ibn al-Salaah al-Shaafa’i in his fatwas.
Shaykh al-Islam (may Allaah be pleased with him) said, after mentioning the two reports concerning this matter: every aspersion against a person’s honour, whether it is truthful or is a lie, comes under the heading of qadhaf (slander), because qadhaf may be true, in which case it is gheebah (backbiting), or it may be false, in which case it is buhtaan (lying). Our companions favoured the view that the victim should not be told, rather du’aa’ should be offered for him, because telling him would cause more hurt and annoyance, and if a person is harmed by knowing what has been said about him, that causes more harm than if he does not know. Telling him leads to these two evils and there is a third evil – even if it is true – which is the destruction of any friendship that may have existed between them, or the renewal of enmity and hatred. Allaah has enjoined us to unite and not to divide. This evil may be worse in some cases than in others.
Those scholars who said that it is essential to seek the forgiveness of the one who has been wronged in this matter exempted the backbiter from having to do that if the victim is dead or absent. In that case it is sufficient to pray for forgiveness for him, make du’aa’ for him and do a lot of good deeds.
See al-Adhkaar by al-Nawawi, p. 308
And Allaah knows best.