Bring ginger, sugar, and 2 cups water to a boil in a small saucepan. Remove from heat, cover, and let sit 30 minutes. Strain syrup through a fine-mesh sieve into a bowl or jar and discard ginger. Let cool completely. (You should have about 2 cups.)
For each drink, add ¼ cup ginger syrup and ½ cup club soda to an Old Fashioned glass filled with ice; stir to combine. Garnish with a lime wedge.
DO AHEAD: Ginger syrup can be made 1 month ahead. Cover and chill.
