Put the grated ginger in a saucepan or pot with a few mint leaves that you have "muddled" by mashing or pressing them to release the minty aroma.
Add the water and agave and simmer the mixture for 5 minutes, stirring occasionally.
Strain the liquid into a heatproof cup and let it cool to room temperature, either on the counter or in the refrigerator. In the latter case it should be cool enough in about 10-15 minutes.
During the simmering, the liquid reduces by about a half, so you will have 2 ounces left. Pour it into a glass, add a few ice cubes and 6 ounces of soda, garnish as you please and enjoy!