In a metal cup, add 5 of the blackberries, 2 of the lime wedges, the mint leaves and brown sugar. Muddle with the end of a wooden spoon, pressing the sugar into the lime and mint leaves to release the juice and mint oil.
Add the rum to the metal cup, and muddle some more until the sugar has dissolved.
Pour the chilled club soda into the cup and gently swirl.
Pour the blackberry mojito mixture over chilled ice. Top with more club soda to reach the top of the glass if needed.
Garnish with the remaining blackberries and lime wedge and a sprig of mint if desired. Serve immediately.