Bring the water to a boil and pour it over the lemon verbena or mint leaves in a bowl. Cover and let steep for 20 to 30 minutes, then discard the leaves.
Cut the melon open and scrape out the seeds, but don't discard the tender, pale flesh surrounding the seeds--it's so sweet and tasty. Remove the rinds and cut the melon into chunks. You want about 4 cups chopped melon.
Using a standing blender or food processor, blend the verbena infusion, melon chunks, lime juice and sugar. (I like to do a couple of batches in my blender.) Pour the puree into a large bowl and give it a stir. Taste the pale green deliciousness and feel free to add a little more agave or sugar if your melon isn't super sweet, keeping in mind that sweetness diminishes with freezing. Your puree can be made up to 24 hours before you want to freeze it. I usually let it chill in the fridge overnight.
Pour the puree into a shallow baking dish. Cover and freeze for 45 minutes to 1 hour. Remove the dish from the freezer and stir with a fork to break up any frozen bits. Make sure to scrape the edges of the pan--this is where freezing begins. Continue to freeze in 45-minute intervals, interspersed with scraping and freezing again. After a few hours of freezing and scraping your granita should be ready to eat. (If you forget to stir and the granita solidifies, don't worry; you can use a fork to scrape frozen granita and the texture will still be fantastic.)
The granita keeps in the freezer for 2 weeks.