Make the blueberry syrup:  Place the blueberries and water into a medium pot and bring the mixture to a boil over high heat. Reduce the heat to low and simmer for 20 minutes, stirring often. While the mixture is still hot, carefully strain the mixture through a sieve  and into a heatproof bowl. Reserve the strained liquid and discard any solids that have been left behind in the strainer. Measure 1 cup of the warm strained blueberry liquid and place it into a bowl. Immediately add the sugar. Stir vigorously until the sugar is dissolved. Set aside.

Make the cocktail: Place the rum, blueberry syrup, lemon juice into a shaker with ice and shake vigorously for 30 seconds. Pour the cocktail up into a coupe and garnish with a skewer with a few fresh blueberries and a lemon slice. Serve immediately.