Squeeze 4 lime wedges into a highball or Collins glass, then drop wedges into glass. Add blueberries, 3 level bar spoons of sugar, and 1 ounce club soda to glass.
Use a muddler to press and twist ingredients together, mashing the blueberries as you do. There should still be chunks when you're done; do not pulverize anything.
Place mint leaves in the palm of your hand, then smack leaves with your other hand 5 to 6 times, or until fragrant.
Rub leaves on glass rim, then drop into drink.
Add rum to drink, then fill glass with ice. Top off with club soda.
Use a bar spoon to stir ingredients vigorously until drink is well mixed.
Serve drink immediately. If desired, garnish with a lime wedge, sprig of mint, and fresh blueberries. Enjoy!