Cut orange in half. Cut one half into wedges; wrap and chill overnight. Juice the remaining half; set juice aside.
In ablendercombine honeydew melon, tequila, orange liqueur, and the reserved orange juice; cover and blend until smooth. With blender running, add ice cubes, a few at a time, blending until mixture becomes slushy. Pour into a 2-quart freezer container. Cover; freeze overnight.
To serve, rub the reserved orange wedges around rims of eight glasses. Dip rims in coarse sugar or black sea salt to coat; set aside. Using a heavy spoon, scrape frozen mixture; scoop mixture into prepared glasses. Add 1/4 cup of the chilled lemon-lime beverage to each glass. Drizzle with grenadine.