Bring 3 1/2 cups water to a boil with 1/3 cup coarsely grated peeled fresh ginger and 3/4 cup sugar; stir to dissolve the sugar. Remove from the heat, add 3/4 cup dried hibiscus flowers and steep 6 minutes. Strain through a fine-mesh sieve into a pitcher, gently pressing on the solids. Stir in 3 cups cold water, 1 cup dark rum and 1/3 cup fresh lemon juice. Chill. Serve over ice.