He borrowed some gold from his wife, and when the time came to pay her back, the price of gold had risen. Does this come under the heading of riba?

If someone borrows gold, then he must pay back in gold of the same weight, and the difference in value of the gold makes no difference. The increase in its value is not regarded as riba at all.

It is permissible for you to give cash instead of gold, but that is on condition that it be equivalent to the price of gold on the day of paying off the debt, and you should pay the price in full, and not leave anything of what you owe outstanding.

The scholars of the Permanent Committee said:

Loans must be paid off in the same type of wealth, and this is what fairness dictates. The rise and fall in value is something that may benefit or harm either party. Fluctuation in prices happened at the time of the Prophet (blessings and peace of Allah be upon him), but the basic Islamic principle that was ordained for the Muslims was not changed because of that, and they are to handle their affairs in the light of that ruling.

The borrower may return the value of the loan at the time of paying off his debt, if the lender agrees to that, because it is soundly narrated from Ibn ‘Umar (may Allah be pleased with him) that he said: We used to sell camels for dinars and accept dirhams, and we used to sell them for dirhams and accept dinars. The Prophet (blessings and peace of Allah be upon him) said: “There is nothing wrong with you accepting them at that day’s price, so long as you do not part with any outstanding business remaining between you.”(Fatawa al-Lajnah ad-Da’imah 13/275).

They also said:

There is nothing wrong with borrowing gold then returning the equivalent amount, because the Prophet (blessings and peace of Allah be upon him) said: “Gold for gold, weight for weight, like for like.” Similarly, he may return its value at the current price, hand to hand."(Fatawa al-Lajnah ad-Da’imah  14/113).

Conclusion: what you must do is return the gold as you took it, by weight, and the increase in its price is not regarded as riba.

And Allah knows best.