Bring the sugar and 3/4 cup water to a simmer in a small saucepan over medium heat, stirring until the sugar is dissolved. Remove from the heat and add the mint, pressing with a wooden spoon to submerge. Let cool completely, then remove the mint with a slotted spoon and discard.
Combine the mint syrup, honeydew and lime juice in a blender and puree until smooth. Pour into a 7-by-11-inch or 8-inch square metal baking dish.
Freeze the honeydew-mint mixture until ice crystals begin to form around the edges, about 45 minutes. Scrape the crystals toward the center of the pan with a fork, breaking apart any large chunks. Continue freezing, scraping every 30 minutes or so, until all of the granita has been scraped into flakes, 3 to 4 hours. Spoon into chilled bowls and top with mint leaves.