To make the simple syrup, combine all of the ingredients in a small saucepan. Bring the mixture to a boil, then reduce heat to low and simmer for 15 minutes or until sugar is fully dissolved. Cool completely.
To make the drinks, place 3 Tablespoons of the simple syrup plus the mint leaves in the bottom of a glass. I like to include some of the whole berries in the drink, but feel free to strain them out if you prefer. Using a muddler or the end of a wooden spoon, muddle the mint and blueberries together in the bottom of the glass.
Add the juice of ½ a lime, enough ice to fill the glass, plus the club soda to top it off. Garnish each glass with a lime wedge or slice and a sprig of mint. Stir and serve immediately.