Piece Quests can be attempted up to three times a day. You may use Lodestar Gems to attempt these Quests up to two more times on the same day.
To recover the number of times you can take on Piece Quests, tap the dimming "Pieces" icon on Piece Quests screen, and then tap "Next" > "Recover".
The first time requires 20 Lodestar Gems, and the second time requires 30 Lodestar Gems.