Bring blueberries, water, sugar, and ginger to a boil in a saucepan over medium-high heat. Reduce heat to a simmer and cook for 5 minutes, crushing the blueberries occasionally with a masher. Remove from heat and let cool, about 10 minutes. Pass the mixture through a food mill or fine mesh sieve to remove the skins of the blueberries. Set aside 1/4 cup syrup and store the remainder in the refrigerator for up to 1 week.