In a large saucepan, bring water to a boil; remove from heat. Add mint leaves and tea bags; steep, covered, 3-5 minutes according to taste, stirring occasionally. Discard mint and tea bags. Stir in sugar.
Place 2-1/2 cups honeydew, 2 cups tea and 1-1/2 cups ice in a blender; cover and process until blended. Serve over additional ice. Repeat with remaining ingredients.