Peel the ginger. Using a spoon or vegetable peeler, peel the fresh ginger, then finely grate it.
Heat the water. Bring the water to a boil in a saucepan.
Infuse the ginger. Remove the pan from the heat, add the grated ginger, and let it steep for 10 minutes to 1 hour, depending on how strong you want the ginger flavor to be. The longer it steeps, the more intense and spicy the drink will be.
Strain the ginger juice. Pour the mixture through a fine-mesh strainer into a clean bottle or jar, pressing gently on the grated ginger to extract as much liquid as possible. Let it cool, then cover and refrigerate.
Serve. Pour the ginger juice into a glass and add fresh lemon juice or a few lemon slices. Sweeten with a little honey or maple syrup, if desired. For a little extra heat, add a small pinch of cayenne pepper.
Chill and enjoy. Add a few ice cubes and serve cold.
Notes
The longer the ginger steeps, the stronger and spicier the flavor will be. For a milder ginger juice, 10 to 15 minutes is plenty; if you like a more intense ginger kick, let it steep for up to 1 hour.