Peel and chop the cucumber. Remove seeds if desired.
Cut the honeydew melon into 1-inch cubes.
Add melon, cucumber, mint leaves, lime juice, honey, and water to the blender.
Blend until smooth, about 30-45 seconds.
Strain the mixture through a fine-mesh strainer into a pitcher.
Fill tall glasses with ice cubes.
Pour the strained mixture into glasses, filling them 3/4 full.
Top each glass with club soda.
Garnish with cucumber ribbons and mint sprigs.