Add the cucumber and 1/4 cup water to the bowl of a food processor. Process until the cucumber is liquified.
Make lime simple syrup by combining sugar, lime zest, and remaining water in a small saucepan.Heat until sugar has melted. Chill until ready to serve.
To make the drink: place 2 basil leaves, 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.