In a mini food processor, pulse the sugar and the mint leaves until the mint is finely chopped and the sugar is fragrant. Place the mint sugar into a saucepan with the water and bring to a boil to dissolve. Boil the syrup for 60 seconds, remove from heat, and strain through a fine mesh sieve.
Place the honeydew in a large bowl (or a blender, you may want to do this in batches if using a blender), along with the lime juice and mint syrup. Pulse with an immersion blender until smooth and no chunks of honeydew remain. Divide the mixture between popsicle molds, freeze 30-60 minutes, insert sticks, and continue to freeze until solid (8 hours is best).