Combine the sugar and water in a medium saucepan. Bring to a simmer. Cook 3-5 minutes or until the sugar has dissolved. Cool completely.
Add 2-4 mint leaves to the bottom of a sturdy glass. Squeeze 1 lime wedge on top of the leaves. Drop the wedge into the glass. Muddle until fragrant.
Fill the glass with ice (on top of the lime and mint).
Top with 2 tablespoons of the simple syrup and 1.5 oz of rum. Top with club soda and enjoy.