Mix the lime juice and sugar in a small bowl. Put the grated ginger in a small strainer. Lower the bottom of the strainer into the sweetened lime juice and stir the ginger around with a spoon while it is still in the strainer. Lift up the strainer and discard any remaining ginger pulp. Divide the ginger-flavored lime concentrate into two tall glasses. Add half of the water to each, along with some ice cubes, and stir vigorously.