In a cocktail shaker, add the rum, lime juice, simple syrup, and mint leaves.
Muddle (smash) the mint leaves well.
Add ice and shake the cocktail, then strain into a tall glass such as a Collins or Highball glass.
Add ice and top off with the club soda.
Garnish with a mint sprig.
Notes
You can buy simple syrup or make your own. The simplest way is to combine equal parts water and sugar in a small pot over medium heat. Whisk until the sugar is fully dissolved, and do not allow the mixture to boil. Cool and use as needed, or store in a container in the refrigerator and use for months as needed.You can use any rum, but white rum is traditional.You can use plain club soda or sparkling water instead of lime-flavored, but I think lime-flavored is nice. Also, if you do not have simple syrup you could use a sweetened sparkling water as the sweetener in this cocktail.