Make ginger syrup: In a small saucepan, combine honey, sliced ginger, and 1 cup of water. Bring to a gentle simmer over medium heat, stirring until honey dissolves completely. Remove from heat and let steep for 5 minutes to infuse the ginger flavor.
Strain the mixture through a fine-mesh strainer into a heat-proof bowl, pressing on the ginger slices to extract maximum flavor. Discard the ginger pieces and allow the syrup to cool for about 5 minutes.
Juice the limes while the syrup cools. Make sure to roll the limes firmly under your palm on the countertop before cutting—this helps release more juice.
Combine all components in a large pitcher: pour in the lime juice, ginger-honey syrup, and remaining 3 cups of cold water. Stir thoroughly to incorporate all ingredients.
Taste and adjust as needed—add more honey if you prefer it sweeter or more lime juice for extra tanginess. Remember that once ice is added, the flavor will dilute slightly.
Chill thoroughly in the refrigerator for at least 30 minutes before serving, or serve immediately over ice for instant gratification.
Present beautifully by garnishing each glass with lime slices and sprigs of fresh mint if desired.