In a blended combine the honeydew, pineapple, 1 cup of water, mint, lime juice, and ¼ cup of sugar. Blend until smooth.
Pour the mixture through a fine mesh strainer allowing the liquid to go into a large bowl. Swirl around the solid parts and press until most of the liquid is out.
Scoop the fruit solids back into the blender with 1 cup of water. Blend until smooth. Pour the mixture through a fine mesh strainer allowing the liquid to go into a large bowl. Swirl around the solid parts and press until most of the liquid is out. Discard solids.
Pour the liquid into a large pitcher. Add remaining water and sugar and mix well.