When you find a treasure chest with a Might Crystal at the Great Fairy's Fountain, the Great Fairy offers an artifact that detects such crystals... for a price.
Where to Start the Quest
Go to the Great Fairy's Fountain, which is near the middle of Lake Hylia. You will find Lake Hylia southeast of Hyrule Castle Town. After you reach a certain point in the game, there will be a treasure chest in front of the fountain. Open it and there will be a cutscene, which starts this quest.
How to Complete the Quest
Go to Gerudo Town in southwest Hyrule and visit the shop there. Talk to the shopkeeper. She says to find a floral seashell and a magma stone. She says to ask around Sea Zora Village about floral seashells, and ask at the Goron City shop about magma stones.
Floral Seashell
In the Sea Zora village, go to the village shop (to the left of the entrance) and talk to the shopkeeper. You need to bring an Unfortunate Smoothie. If you don't have one, see the list of Unfortunate Smoothie combinations, and go to a Smoothie Shop to make one. There is a Smoothie Shop to the north of Sea Zora Village.
After you have an Unfortunate Smoothie, talk to the Zora shopkeeper again. There will be a cutscene.
Magma Stone
In the Goron City shop, talk to the Goron with the exclamation mark above his head. Then go to the Lizalfos Burrow and defeat all the Lizalfos there. Afterward, there will be a cutscene.
Visit the Gerudo Artisan
Talk to the Gerudo artisan and there will be a cutscene.
Visit the Great Fairy
Go back and talk to the Great Fairy.
Quest Reward
You receive the Might Bell, which helps you find Might Crystals when you equip it.