Prepare the honeydew syrup: Combine the honeydew cubes, sugar, and water in a small saucepan. Bring the mixture to a gentle simmer over medium heat, stirring until the sugar dissolves completely (about 5 minutes).

Infuse the syrup: Once simmering, remove the saucepan from the heat. Add the 10 mint leaves to the syrup mixture, press them down lightly, and let steep for 15 minutes.

Strain and cool the syrup: Strain the syrup through a fine-mesh sieve into a clean container, pressing gently on the solids to extract all liquid. Discard the solids. Allow the syrup to cool completely in the refrigerator (at least 30 minutes).

Assemble the mocktails: Divide the cooled honeydew syrup evenly among four tall glasses filled with crushed or cubed ice. Add 1 tablespoon of fresh lime juice to each glass if desired.

Top with effervescence: Top each glass with approximately 6 ounces of chilled sparkling water or club soda. Stir gently to combine.

Garnish and serve immediately: Garnish each mocktail with a fresh mint sprig (gently clap the sprig between your palms before inserting to release its aroma) and a lime wheel on the rim.