First make the mint simple syrup: Put 1 cup of sugar and 1 cup of water into a saucepan over medium-high heat. Bring to a boil, stirring occasionally until the sugar is dissolved. Turn off the heat. Bruise the mint a little by rubbing it between your fingers. Add mint to the syrup and set aside until it cools, at least 1 hour. Strain into a clean jar and discard the mint. Cover the jar with a lid. Leftover syrup can be stored in the refrigerator for up to 2 months.
To make the drinks, fill 4 glasses with ice. Put 2 tablespoons of mint simple syrup into each glass. Top with about a cup of club soda and a teaspoon of lemon juice. Stir gently and garnish with a mint sprig. Add 1 1/2 ounces of gin to each glass, if desired.