Combine water, sugar, and lemon verbena leaves in a small saucepan. Bring to a boil, stirring until sugar dissolves. Simmer for 1 minute, then remove from heat and let syrup steep for about 30 minutes.
Strain syrup through a mesh strainer into a glass jar and let cool completely. To store, keep refrigerated for up to one month.
For the Slushies
Place honeydew melon, lemon juice, simple syrup, vodka, and ice into a blender. Pulse to combine; the consistency should be similar to a frozen margarita. Add more water or ice if necessary.