In a sturdy Highball or Collins glass, combine the blueberries, lime juice, and sugar. Use a muddler or the back of a spoon to gently crush the blueberries and mint together.
Add the fresh mint leaves, taking a moment to clap them between your hands before dropping them in. Pour in the white rum and stir gently to combine the flavors without bruising the mint too much.
Fill the glass with ice cubes to the top. Fill the glass with club soda. Stir gently with a long spoon.
Garnish your mojito with a few extra blueberries, a sprig of mint, and a lime wheel on the rim of the glass. Serve with a straw.