Add the blackberries, simple syrup and lemon juice to a cocktail shaker. Use a wooden spoon, or a muddler, to smash the blackberries and release the juice.
Add the ice cubes and rum to the shaker. Add the lid to the shaker and shake vigorously for 45-60 seconds.
Strain the blackberry cocktail into 2 old fashion style glasses. Add more ice, garnish with a lemon or lime. Serve immediately.