In a high-powered blender, combine the honeydew and 1 cup water and blend until completely smooth (see Cook's Note). Transfer to a pitcher and stir in the Mint Simple Syrup and lime juice. Add more water or seltzer to taste.
Mint Simple Syrup:
In a small pot over medium heat, add the sugar and 1 cup water and cook, stirring, until the sugar is dissolved. Remove from the heat, add the mint and let steep for 5 to 10 minutes. Strain out the mint and refrigerate for up to 2 weeks.
If using a normal blender, strain with cheesecloth after blending.