Cut the honeydew into small cubes, removing the rind and seeds.
Add the honeydew cubes to a blender and blend until smooth.
Strain the melon puree through a fine-mesh strainer into a cocktail shaker.
Add rum, lime juice, simple syrup, and ice to the shaker.
Shake well for 15-20 seconds until thoroughly chilled.
Strain the mixture into collins glasses filled with ice.
Top with club soda and stir gently.
Garnish with lime wheels and melon balls.
Notes
Store extra honeydew puree in an airtight container in the fridge for up to 3 days.Choose ripe honeydew that feels heavy and has a sweet aroma near the stem end.The drink can be made without alcohol by replacing rum with coconut water.Add mint leaves for extra freshness and color.