Electronic – How do the lights on stairs work

The "sometimes stays on" action is what is called "hysteresis". This is designed to stop the light flashing on and off when it is on the verge of night or day.
Once it is on it adjusts its trigger level point such that it needs to get somewhat lighter befre it will go off again. Once it is off it adjusts its trigger level point such that it needs to get somewhat darker lighter before it will go on again.

When you adjust the conditions to be "just right" you dro it into a "twiligh zone" (literally in this case" where it needs more dark or more light to cause action.

Here is a typical nightlight control circuit with hysteresis.
The ORP12 at R8 is a light variable resistance.
R7 is a sensitivity control.
As light level varies the voltage at the R8/R1 boundary varies. The opamp compares this voltage with the voltage from R2 / R3 divider and switches when the ORP8 controlled voltage transitions past that voltage i either direction.

Hysteresis is meant to be added by voltage from the output of the opamp, via R4, back to its input. The interesting this is that they have connected it to th wrong plac !!! :). R4 should connect to opamp pin 3 = non inverting input and not to pin 2 = inverting input. When it is connected to the + input, anything that happens at the output is fed back as positive feedback and enhances the action. This exact circuit diagram (same drawing without change) can be found in many locations on the internet with some mentioning hysteresis - which it has not got.

So - this is an example of the sort of cct that may be in your light - but they sound like they have connected it correctly in your case.

This one does it correctly but its action is slightly harder to follow.
Circuit from here with an OK discussion. He specifically notes R5 R6 & R7 as being involved in setting the amount of hysteresis, which they are. Bigger version of diagram here

Circuitry in box at top left varies the "attack & decay" rates of the LDR (ligh dependant resistor) effect - see his text for why.

There is a couple of methods commonly used in your situation.

1) (not recommended) Use the protective Earth ground connection that should be present in your box instead of a Neutral connection.

Note that although this works quite well from a technical perspective, it won't pass UL or CSA certification (nor most other certifying authorities). One reason is that if the Earth connection should become disconnected or broken, you are deliberately passing current down the Earth conductor, which would cause it to become live in the event of a disconnected Earth.

2) Do some fancy phase control during the ON state.

Stealing power while the switch is OFF (open) is easy - you treat the Switched conductor as if it were the Neutral. That is: the load provides the continuity to the Neutral conductor. You will have approximately the full AC mains voltage across the switch.

This is (usually) OK so long as you are stealing only a few mA. However, there are some loads that this won't work with - I'm thinking of the little Egg-shaped strobe lights that used to be in discos in decades past. There may be other loads that aren't happy with this as well.

Stealing power when the switch is ON (closed) is a little more difficult. What you do is use a triac as the switching element and delay turning the triac on until the voltage across the triac is the desired supply voltage that you are trying to achieve. A diode picks that voltage off and stores it in a bulk storage capacitor. Then turn the triac ON.

Modern X10 light switches and some other smart light dimmers work this way.