In a small saucepan, combine 1/2 cup (120 ml) water, the agave, and the chopped fresh ginger. Bring to a simmer over medium heat. Once simmering, cook for 10 minutes to extract the ginger flavor. Remove from the heat.
Meanwhile, to a glass jar or pitcher add lemon juice and remaining water. Place a strainer over the jar/lemon mixture and pass the ginger syrup through it, removing any small pieces of ginger. Whisk the lemonade well to mix in the ginger syrup, and serve it over ice. Optionally, garnish with fresh mint leaves. Serve on its own or as a mixer with tequila, gin, or vodka.
Lemonade will keep in a sealed container in the refrigerator up to 5 days, or freeze in popsicle molds for a sweet, cold treat! Can also be frozen into ice cubes and added to summer beverages.