Her mother-in-law gave her some gold at her wedding party, and is demanding it now, claiming that it was a loan

Praise be to Allah.

Firstly:

If your mother gave the gold by way of a loan to be returned, and your wife said: No, rather it is a gift or a present, then your mother has no right to the gold unless she swears that it was given as a loan.

One of the fundamental principles of jurisprudence is that if the recipient and the giver differ concerning what was given by one to the other, then what counts is what the giver says.

This was stated by az-Zarkashi in al-Manthur fi’l-Qawa‘id (1/145), where he gave as an example of that:

If he gives some money to his wife, and says: I gave it to her as part of the dowry, but she says: No, rather it is a gift, then what counts is the word of the giver. Ar-Rafi‘i narrated that in Kitab as-Sulh from his companions. And he said in Kitab as-Sadaq: If the spouses differ regarding receipt of money, and he says: I gave it as dowry, and she says: Rather it was a gift, then what counts is his word, if it is supported by his oath. End quote.

It says in Hashiyat Ibn ‘Abidin (5/710): A man bought some jewellery and gave it to his wife, who used it, then she died. Then the husband and her heirs disagreed as to whether it was a gift or a loan. What counts is the word of the husband, supported by his oath attesting that he gave it to her as a loan, because he denies that it was a gift. End quote.

See also: al-Fatawa al-Hindiyyah (4/399).

Secondly:

You should explain to your mother that taking back a gift is haram, because of the report narrated by Abu Dawud (3539), at-Tirmidhi (2132), an-Nasa’i (3690) and Ibn Majah (2377) from Ibn ‘Umar and Ibn ‘Abbas (may Allah be pleased with them) from the Prophet (blessings and peace of Allah be upon him), who said: “It is not permissible for a man to give a gift or a present then take it back, except in the case of what a father gives to his son. The likeness of the one who gives a gift then takes it back is that of a dog that eats then when it is full it vomits, then it goes back to its vomit.” This hadith was classed as sahih by al-Albani in Sahih Abi Dawud.

And Allah knows best.