1. Freeze cubed honeydew at least 3 hours.
2. Place 2 cups frozen honeydew, 1 cup lime juice, 1 cup sparkling water, 1/2 cup sugar, and 1/4 cup mint in a blender; process until smooth. Repeat procedure with remaining honeydew, lime juice, sparkling water, sugar, and mint.
3. Pour 1/2 cup ginger ale in bottom of each of 8 chilled glasses. Pour 1 cup honeydew mixture in each glass; stir gently. Garnish with lime slices, if desired.