Make the simple syrup: Combine the sugar and water in a small saucepan. Bring to a boil and let simmer for 3 minutes, stirring occasionally until all of the sugar is dissolved. Remove from heat and let cool completely before use. Set aside 4 tbsp. Remaining simple syrup can keep refrigerated for up to weeks. Make the agua fresca: fill a blender or food processor with the cucumbers, water, and the fresh ginger. Blend or process on high, repeating until all of the cucumber has been puréed. Pour the mixture through a fine strainer and discard all of the solids. Stir in 4 Tbsp. of the simple syrup and fresh lime juice. Serve over ice.