In a blender, combine blueberries and powdered sweetener. Blend to a thick puree. Transfer to a pitcher.
Tear mint leaves by hand and add to a pitcher that holds at least one liter. Crush with a muddler or the end of a wooden spoon to release the oils. Add blueberry puree, white rum, lime juice and soda and stir to combine.
Fill 4 highball or bucket glasses with crushed ice. Pour mojitos into glasses, garnish with blueberries and mint leaves and serve.