To make simple syrup: In a saucepan, bring 1/4 cup water to a simmer. Add 1/4 cup sugar and stir until sugar dissolves. Remove from heat and let cool.
In a pitcher, muddle mint leaves, blackberries, and blueberries with a wooden spoon. Squeeze in lime juice and add wedges. Add simple syrup and rum. Stir until combined. Add club soda. Refrigerate, then pour over ice when ready to serve. Garnish with a mint leaf and additional lime wedge.