Put the honeydew, lime juice, basil syrup, tequila, and ice cubes in a blender and mix until well incorporated, about 30 seconds. Pour into a margarita glass and serve garnished with fresh basil and a slice of honeydew.
In a 1-quart saucepan, combine the sugar and water. Put it over medium heat and stir to help the sugar dissolve. Once the sugar has dissolved and the syrup is hot, take it off the heat and add the chopped basil. Allow the syrup to cool for 15 minutes, strain it through a fine mesh sieve, and discard the basil. The syrup may be made up to 2 weeks in advance and refrigerated in an airtight container until ready to use.