There are three different quests related to colossal fossils in The Legend of Zelda: Tears of the Kingdom. You will get the first quest from Loone to find the Fossil around Death Mountain. Finding the fossils is easy as they will be marked on the map but each of them will have some kind of puzzle for you to solve.
Each quest will give you 50 Rupees, meaning you can land a nice 150 easy Rupees for completing this line of quest. This guide will cover all of the Colossal Fossil quests and cover how to complete each of them.
Eldins Colossal Fossil
When you arrive at the skeleton of Eldin you will find Loone standing near the front of the head. She will tell you that the skeleton is missing an eye and will want you to fix it.
Walk underneath the head of the statue and you will find a pile of breakable boulders. You can smash all of these and when enough are gone, you will find a large white ball. This is the eye. Use the ultrahand to bring the eye outside and put it into the eye socket on the left side where the eye is missing.
Return to Loone and she will give you a 50 rupee reward for restoring the skeleton.
Hebra Colossal Fossil
The Hebra fossil can be bound in the snowy mountain region. You will need to go into an underground cave. When you speak with Loone, she will say that the skeleton is missing a fin bone and a chunk of the spine and will ask you to fix it.
The fin bone can be found to the left, sticking out of a wall that has some more breakable boulders. Destroy these and you can use the ultrahand to carry it back to the fin and put it into place. It will snap into place once you have it the right way.
The spine piece is frozen inside a large block of ice. You can use a fire arrow, bomb or anything that generates heat or fire to melt the ice. Using the ultrahand again, you can put the piece of the spine into the correct location. This piece is quite tricky to set into position. I had to move around a few locations, pulling the piece back and fourth until it finally clicked into place. If you are having trouble, just keep rotating it until the little fins match up as close as possible with the rest of the spine.
Gerudos Colossal Fossil
This quest will have you needing to travel to the western side of the Gerudo Desert. This will also require you to go down into a cave. When you speak to Loone, she will talk about the body of the baby but it is missing some pieces and wants you to restore the skeleton.
You will find the tail inside a pile of sand beneath the main fossil. Use one of the fans nearby to blow the sand away and reveal the tail. Bring this back toward the head.
The body is actually stuck in the roof. You will need to use a bomb arrow or clumb up onto the head of the parent and try bash away enough of the rocks to make the body of the baby fall to the ground.
Using the ultrahand, try to repair the body and it will take a little work to get right but try to make sure the tail lines up correctly with the spine on the body and then make sure the head is in the correct spot. You will know you have it right when you see the red circle show up when you look toward Loone. Speak with her and she will give you the final 50 Rupees for this quest.