This page explains how to start and complete the The Jewel Trade side quest.
How to Start the Quest
First, you need to complete the Divine Beast Vah Rudania Dungeon. Then you can talk to Ramella, a Gerudo in Goron City, to start the quest.
How to Complete the Quest
Sell Ramella 10 of the requested ore and she will pay more for it than normal. From now on, she will request different ores and you can sell them to her for more than normal if you have them.