To make the blueberry simple syrup, bring the water, granulated sugar, blueberries and ginger in a medium sauce pan. Boil until all sugar has dissolved, about 3 minutes. Strain mixture to remove pieces of blueberry and ginger. Refrigerate simple syrup until ready to make the cockatils.
To make the mojito, place the blueberries and mint in a cocktail shaker and muddle by lightly pressing down and twisting to release mint flavors. Add the simple syrup, white rum and lime juice and shake vigorously, about 1 minutes.
Pour mixture over ice and top with a splash of club soda. Garnish with blueberries, mint and lime.