In a pot, heat water and sugar until boiling. Add 5 mint leaves, reduce to a simmer and cook 5 minutes. Set aside to cool, allowing mint to steep. Strain out mint when mixture has cooled.
To make individual drinks: Place 1/3 cup of blackberries in a glass with 1/2 tablespoon of lime juice and smash with a spoon. Add 1/2 cup of the cooled mint syrup and top with 1/2 cup of sparkling water or club soda. Add a few ice cubes, garnish with a mint leaf and serve immediately.