Bring a large pot of water to a boil. Slowly add desired amount of tapioca pearls. Bring water back up to a simmer and cook about 20 to 30 minutes, stirring occasionally so pearls don't clump together. Rinse pearls in cold water once cooked.
In a food processor, pulse honeydew until completely blended. Using a fine mesh strainer with a bowl underneath, strain the pulp from the honeydew mixture. (It should yield about 2 cups honeydew juice.) Tear mint leaves and add to the bottom of a glass. Add tapioca pearls, ice, honeydew mixture and a splash of coconut water. Stir and enjoy!