Late at night, however, they flicker dramatically, sometimes going out completely for a full second or more, then blinking on again. If I shine a small torch at them, they revert to their steady daytime glow. Putting two lights in adjacent sockets so that one is lit by the other does not stop the flickering. A neon glow lamp glows when the voltage across the electrodes exceeds the breakdown voltage of the rarefied gas inside. The current flow ionises the gas.
Ions then recombine to create light, the colour depending on the kind of gas.
If the mains voltage is varying because of the demand for power, the peak voltage may not be enough to ionise the gas, or the current may only flow one way through the gas, causing erratic behaviour. Shining a light onto the electrodes excites the atoms on the surface enough to allow them to release electrons at a lower voltage. The shorter the wavelength of light the better the effect. The questioner could try holding an old watch with a radium dial near the erratic neon lamp to observe this. The reason having two lamps doesn't help is that both lamps turn off twice during each AC cycle.