Change of potential difference during electrolysis while adding salt
Tungsten lamps change resistance as they warm up. At room temperature, the resistance of a small flashlight lamp (e.g. #222) might be 10 ohms when brightly lit, but only 1.5 ohms at room temperature.

At first, the tungsten filament acts like a short circuit, so most of the voltage of the battery is dropped across the series resistance of the electrolytic cell. Once enough current goes through the lamp, though, its resistance abruptly rises and more of the voltage is dropped across the filament.

If you use an LED instead of a tungsten-filament bulb, the change in voltage would be more linear as you add salt. Note that an LED will be effectively an open circuit until a minimum voltage is reached (~1.6 V for a red LED, ~3 V for blue or white LED), but then current increases linearly until ~20-50 mA, at which point some may burn out.