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What does the neutral wire do on a light switch that responds to wifi?

My suspicion is that when the switch is in the off position, the electronics in the switch need to still be powered, so the neutral wire allows a tiny constant current through the switch so it can respond to a wifi instruction to actuate the switch.

If this is so, could the extra neutral wire going to the switch be a very light gauge wire?

Because unlike a normal switch, the wifi switch needs power, so it needs a neutral wire to complete a circuit at the switch between the unswitched hot wire and neutral for that power.

The neutral wire to the switch cannot be a light-gauge wire (at least if you are in an area subject to NEC codes; likely similar elsewhere in the world) because codes are based on what the circuit breaker protecting the circuit can supply when choosing wire size, so for 15A circuits the wire must be 14 Ga or larger, and for 20A circuits the wire must be 12Ga or larger. Smaller wires could overheat to the point of causing fires (without tripping the breaker) in the event of a fault condition, even if the normal current in them might be a matter of a few milliamperes.

I regard wifi switches (and thermostats) with the baleful gaze that only a network admin can view such things askance with, but I gather that there are units arranged in such a manner that the actual switch/relay can go in the ceiling where the light fixture that the switch loop originates from is, and the switch loop wires can (without adding any) be repurposed to operate the remote switch in the wall, no longer actually running the power for the lights through there. This type of unit will probably be much easier to install than to attempt to add a neutral wire to a switch box without one.

That is correct. In the mains power world, we have 3 wires.

EQUIPMENT SAFETY GROUND/EARTH which is not used for anything ever (except during a fault condition, of course)...

HOT which is what humans consider the "source" of power... and

NEUTRAL which is what humans consider the "power return". Obviously, it's AC, which makes "source" vs "return" rather arbitrary. This can be said: the system has equipotential bonding that, in normal conditions, places neutral quite close to safety earth. That's why it's called neutral.

When you want to power something 24x7, it must be connected between HOT and NEUTRAL. If it is not, it won't have power. Since presumably you want the WiFi switch to respond to WiFi at all times, you need HOT and NEUTRAL to pull that off.

You may be thinking of early dimmers. Those place themselves in series with the light bulb, so they connect between ALWAYS-HOT and SWITCHED-HOT. When the bulb is on, they create a small voltage drop and power themselves that way. When the bulb is off, they leak a small amount of current through the bulb, which works great with incandescents, but badly with most LEDs. It's possible to build a smart switch this way, in which case it does not need neutral. However it is difficult to design, and doesn't play well with most LEDs and CFLs. This can be solved by using 1 incandescent, or by putting a capacitor in parallel with the light bulbs so some power can leak through 24x7.

A very small number of dimmers place themselves in series with the equipotential bonding that keeps neutral near ground. In short, they bootleg neutral off of ground, but do so in very controlled conditions that satisfy Underwriter's Laboratories that it won't be unsafe. Don't do this yourself, unless instructed to by this particular device's UL-approved labeling and instructions.

If you don't want to hunt down a compatible smart switch, you need to replace the entire cable between light switch and light, with a /3 cable (black red white +ground). Then you make black always-hot, red switched-hot, and white neutral, hooking each as appropriate to the smart switch and up in the lamp.

If it is simpler to run power from the service panel, then run a /2 cable from the service panel to the switch, feed the lamp from the switch, and cap off the supply hot+neutral going into the lamp. (or disconnect it from the lamp circuit in all respects).

If conduit is used with individual wires (THHN), and hot and switched-hot are already colored wires, then you can simply add a similar (THHN) white wire to the pipe.

Other than that, you cannot just run an individual wire. I gather you're a DC electronics guy? This is AC, and magnetic fields from AC have a serious kick. So currents must be equal in all cables or conduits so magnetic fields cancel each other out. Which means all related wires must be in the same cable.

You are allowed to run an individual safety ground (i.e. retrofit), because current does not flow over safety ground (except during a fault).

You can't run a tiny wire because we don't wire houses for ideal conditions, we wire them for fault conditions. Fault conditions call for a shorting smart switch that is flowing 50A through that 15A-rated wire. That 15A-rated wire will heat up at a predictable rate, and the breaker will thermal-trip before the wire heats up to a dangerous temperature. Now if your neutral wire is smaller, it will heat up faster.

That's why you would need to size the entire circuit's breaker for that smallest wire. They don't allow <15A circuits in houses because the incremental cost of copper for a larger wire becomes pretty marginal at that size. That, and economies of scale make #14 effectively as cheap as smaller sizes.