In a small saucepan, bring 1 cup blackberries, the sugar, ginger and 1/4 cup water to a boil, stirring to dissolve the sugar. Lower the heat and simmer for 5 minutes. Let cool, then strain, discarding the solids.
In a large pitcher, combine the gin, lemon juice and blackberry-ginger syrup. Fill the pitcher with ice and stir vigorously. Fill 6 tall glasses with ice. Pour the chilled liquid into the glasses, holding back the melted ice in the pitcher. Top off each drink with the club soda, then garnish with blackberries, a lemon slice and mint.