In a large saucepan, bring water to a boil, then remove from heat. Add ginger and steep overnight. Strain mixture through a fine-mesh sieve, pressing ginger to release flavor, then pour into bottle through a funnel. Add lime juice and 1 cup sugar; stir vigorously. Adjust to your taste with remaining sugar. Cover and store at room temperature; let mixture steep (and dregs settle) 24 hours. Decant to a second bottle with a tight-fitting screw top and refrigerate 3 to 5 days.