The Gem of Kappa is made by collecting all nine Kappa Jewels. The jewels can be found in the Lake Mine that can be accessed during the Winter Season. If you're playing More Friends of Mineral Town, the Gem of Kappa is necessary in order to marry Kappa.
The gem's main purpose is to increase your character's stamina. The Gem can be equipped in your active tool slot. When equipped, your character's stamina will gradually increase when your character is outside walking around. Time must be passing, and it must be equipped as your current tool. If you have the Truth Gem, you can see how much stamina you have regained. The Kappa Gem is the only way in game to recover all of your stamina in a day. Although going to the hot springs or drinking medicine will help, they do not restore your stamina fully.