In a large mixing bowl, combine water, soy sauce, brown sugar, honey, ginger, and garlic. Whisk together.
In a small bowl, combine cornstarch and water. Mix well.
Pour the first mixture into a medium saucepan and cook on medium heat for about 5 minutes.
When the sauce begins to boil, add the cornstarch mixture and whisk continuously.
When the teriyaki marinade begins to thicken (about 2 minutes), remove from heat. Allow to cool to room temperature.
In a dish or large freezer bag, coat the chicken in 3/4 of the teriyaki marinade and place in the fridge for at least 1 hour.
Place the chicken drumsticks on the grill and baste with reserved teriyaki sauce.
Cook for 15 minutes, turn, and baste with sauce again.
Cook for an additional 15-20 minutes or until the internal temperature of the chicken reaches 170°F when measured with a meat thermometer.