Thinly slice the ginger. Add the ginger, lemon juice, and lemon quarters to a small saucepan.
Cover the pot, bring to a boil, and immediately lower the temperature. Bring liquid down to a light simmer. Continue simmering covered for about 30 minutes.
Using a fine mesh sieve, strain the ginger water into a large measuring cup (or another container). You should have about 1 cup of liquid left. If you have less, add filtered water until you have one cup.
Rinse out the saucepan, and pour the ginger water back in. Add the sugar or honey, stir over low heat until the sugar or honey dissolves.
Pour the syrup into a glass jar, or funnel into bottes. Store in the fridge and use within two weeks.
Nutrition information assumes sugar instead of honey.