In a half gallon jar, add lemon rind and then the juice of the lemon. Add sugar and the boiling water and swirl to dissolve sugar and release the oils from the lemon rind.
Add cool water, leaving 3-4 inches of headspace in the jar. Make sure you use cool water so the boiling water doesn't kill the ginger bug.
Add ginger bug discard and a handful of fresh berries etc. if using. Top with water if needed to reach the rings of the jar.
Put a lid on and let sit on counter about 24 hours, until the sugar has been "used up" and there's some fizz. Store in the fridge.
Notes
Make sure and keep your ginger bug fed with a bit of sugar daily and more fresh ginger root every few days.
Always use water that is truly filtered. Any sort of chlorine remaining can kill your ginger bug and your ferment won't work.
You can use fresh or frozen berries to flavor the drink.