They found some gold in their house that belonged to a relative, who came back a month later to take it but they had lost it. What do they have to do?

Firstly: 

This gold that you found was a trust (amaanah) that was in your care and that you had to take care of until its owner came to take it. The ruling concerning a trust, if it is lost by the one in whose care it was due to some shortcomings or negligence on his part is that he is liable for it and must compensate its owner; if its loss was not due to any shortcomings or negligence on his part then he is not liable and is not obliged to do anything. 

Ibn Qudaamah (may Allah have mercy on him) said: 

Any item left in a person’s care is a trust (amaanah). If it is lost or destroyed without any negligence on the part of the one to whom it was entrusted, he is not liable regardless of whether any of the property of the trustee was lost along with it or not. This is the view of the majority of scholars.…

But if the trustee neglected it or fell short in taking care of it and it was lost or destroyed, then he is liable and there is no difference of scholarly opinion that we know of, because the one who destroys the property of another is liable for it, as is also the case if he destroys it without it having been entrusted to his care. End quote. 

al-Mughni, 7/280 

So you should have kept this gold in the same safe place where you put your own gold, and you should not have put it with papers that were easily accessible to people. 

Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allah have mercy on him) said: 

Storing gold and silver is not like storing pots and pans. Pots and pans are kept in clear view in the house and are not locked up, but gold and silver are kept in boxes and chests and are locked away. End quote. 

Al-Sharh al-Mumti‘ ‘ala Zaad al-Mustaqni‘, 10/288 

What appears to be the case is that you fell short in taking care of this gold, so you are liable for it. 

The basic principle with regard to liability is that you are liable for replacing it with something similar to what it was on the day it was lost. If it is not possible to replace it with something similar, then you have to pay its value. If there is a piece of gold on the market that is similar to the piece of gold that belonged to your relative, then he should be given the choice of either having you buy it for him or you giving him its price. 

If there is no similar piece of gold in the market, then you have to work out the value of his piece of gold at the time it was lost, then give him its price. 

Shaykh al-‘Uthaymeen (may Allah have mercy on him) said: 

What counts [with working out the value of a lost or destroyed item] is the time when it was lost or destroyed, because that is when it ceased to be in the possession of its owner, i.e., at the time when it was lost or destroyed.

End quote. 

Al-Sharh al-Mumti‘ ‘ala Zaad al-Mustaqni‘, 10/122 

This is what you have to do for this relative, unless he waives his rights in full or in part. 

And Allah knows best.