Peel and seed the cucumber. Add the cucumber and 1/4 cup water to the bowl of a food processor. Process until the cucumber is liquefied.
Make the lime simple syrup by combining sugar, lime zest, and remaining water in a small saucepan. Heat until the sugar has melted. Chill until ready to serve. (The cucumber puree and simple syrup will keep for several days in the refrigerator)
To make the drink: place 2 basil leaves, the juice from 1/2 a lime, and 1 Tbs of your simple syrup in the bottom of a glass. Use the back of a spoon to muddle the basil. Add 2 Tbs cucumber puree and top with 8 oz club soda. Stir gently to mix.
Sit back, Relax, and Enjoy!