For the mixed berry simple syrup: bring water, sugar, blueberries and strawberries to a simmer. Cook for 10 minutes, smashing the berries as they soften. Place a fine mesh strainer over a clean bowl. Strain syrup. Store in an airtight jar in the refrigerator.
In a cocktail shaker, combine mint leaves, lime juice, simple syrup and white rum. Muddle mint leaves. If you don't have a muddler (I don't) use the end of a wooden spoon or something similar to break up the leaves a bit. Add ice to the cocktail shaker and put on lid. Shake about 30 seconds to chill.
Strain through top of the cocktail shaker and pour over ice. Top with club soda. Garnish with additional mint if desired.