In a gallon glass jar or similar container, combine 1 cup starter (reserving the rest for a future batch of soda), fresh ginger, brown sugar, lemon juice, and water.
Stir together well until the sugar is fully dissolved.
Cover the container with a cheesecloth or a tea towel and leave somewhere warm, out of direct sunlight.
Twice a day for the next 5 days stir the mixture well. This is important - if you don't stir it regularly, you risk the fresh ginger on top of the mixture getting moldy. As soon as you see lots of bubbles forming on top or it fizzes when you stir it, proceed to the next step.
Once the soda is fizzy, it is time to bottle it up. Taste the mixture to make sure it is still sweet – if it is quite tart, add a little more sugar.
Sieve out the ginger, then pour the ginger soda into swing top bottles. Close tightly with the stoppers.
Leave the bottles on the counter for another 24-48 hours to make the mixture more carbonated. Test one bottle by opening the stopper for just a moment. If you hear a 'pop' and see some bubbles, your soda is ready.
Transfer the bottles to the refrigerator and let them chill well before serving.
Consume the ginger soda within the next 14 days or so. It won't spoil if you keep it longer than that, but the bacteria may use up all the sugar by then and turn the soda really tart.