Combine 1 cup water and sugar in a small saucepan over medium-high heat; cook 5 minutes or until sugar dissolves, stirring with a whisk.
Place ginger on a double layer of cheesecloth. Gather edges of cheesecloth; tie securely. Place the cheesecloth bag in a large pitcher, and add lemon juice. Add the sugar mixture and remaining 5 cups water to pitcher, and stir well. Refrigerate the lemonade for 2 hours or until chilled. Discard cheesecloth bag.