Combine the sugar, ginger and 1 cup water in a small saucepan. Bring to a boil, then reduce the heat and simmer until the sugar is completely dissolved. Let cool a bit, strain out the ginger, and then let cool completely.
In a large pitcher, stir together the ginger syrup and vodka. Fill 10 glasses with ice and pour the cocktail over. Top off with a splash of tonic water and garnish with a sprig of mint and a lime wedge.