Peel and grate the ginger and add it, together with the sugar and cold water, to a saucepan. Bring it to the boil and stir until the sugar dissolves. Cover and let steep for 15 minutes. Strain and cool in the refrigerator when done.
Add the blueberries, lime wedges and fresh mint leaves to the glass (or pitcher). Muddle with a wooden spoon so the blueberries are broken and the mint and lime release their juices and flavor. If assembling pitchers you can cover them at this stage and leave in the refrigerator until ready to serve.
When ready to serve, add the simple syrup, vodka and top off with club soda. Give it a quick stir and then pour into glasses with ice cubes. Garnish with a handful of fresh blueberries and a sprig of mint.
The recipe outlined below is for 2 tall glasses, but you can use the ratio to make pitchers (depending on how big your pitcher is double or quadruple the recipe).
Combine the blueberries, mint leaves and lime wedges in the pitcher and muddle it in there with the back of a wooden spoon. When you are ready to serve add the vodka, ginger simple syrup and club soda. Stir and pour into glasses with a couple of ice cubes. Garnish each glass with some fresh blueberries and a sprig of fresh mint.