In a blender, add the melon, lime juice, and ginger. Blend until smooth.
To create an optional decorative rim, run a lime wedge around the rims of your serving glasses. Pour some sugar on a small plate and dip the edge of the glass in the sugar.
Divide the crushed ice between the glasses, and top with the cucumber slices. Pour over top two parts blended melon mixture to one part carbonated water. Add agave syrup to taste, and enjoy!