Place the cucumber and lime juice in a blender and blend until smooth.
Add the cucumber-lime juice and simple syrup to a glass filled with ice, then top with the club soda and stir to combine. Garnish with cucumber and lime slices, and serve.
Notes
Make Ahead: to make this mocktail ahead, blend the lime juice and cucumber, then transfer to a storage container and refrigerate for up to 2-3 days. When ready to serve, pour over ice and top with the club soda.