I put a magnetically ballasted fluorescent desk light on the same table as my PC. But whenever I turn it on or off, only one of my monitors turns off for a few seconds (not the video signal, power related). The weird thing is that, the lamp is plugged in to a different socket than the monitor, and all my monitors' power cables are tied together. None of my other monitors have this problem. Also, the cord of the lamp is not near the monitor cables. Why is this happening?

Isn't somewhere a bad contact, so as you are toggling the lantern switch, it moves some monitor's wiring, causing the connection to be temporarily lost...

No that can't be it. When I press the switch, the monitor is fine. Only when the lamp finally strikes, then it turns off.
I tries shaking the monitor and pressing the switch of the lamp with no tube in it, but it stays on.
Also for clarification, the desk lamp isn't manual preheat, it uses a glow-bottle starter.

And is it really the monitor power shutting down, or it just goes to StandBy/Sleep?
And is it behaving the same way if you disconnect the signal (HDMI, or whatever you are using...) cable towards the computer?
If it is only when the cable is connected, I would check the cable, mainly the shielding/ground connection, the metal connector shells needs to be connected between both ends. If they are not, the problem is bad cable. Very often mainly the cheepeese cables have this connection very poor or even completely missing, so in that case you need some proper cable.
If the cable is OK, then you may check its connection in the monitor/computer: Connect them together by the HDMI cable, disconnect power and all other wiring from both the computer and the monitor and then check continuity (measure resistance) between both schassis. Connect the meter to shielding shells on other connectors, like second input on the monitor and/or USB connector on the computer. Don't rely on the mains power input PE (grounding) contact, as (mainly on the monitor side) it does not have to be really connected to the schassis, often it is used only to connect the shield between primary/secondary of the power supply, but the secondary (and the schassis) remains insulated (cust connected via capacitors; the reason is to prevent large power ground loop currents, yet ensure no path for dangerous voltage may form). If not conductive (more than half Ohm or so) but the cable itself was OK, it could be worth checking the connector soldering: The shield connection may have broken, you need to tie it to a solid ground plane on the PCB.

If everything OK, youmay try to use some clamp-on ferrite beads over the signal cable (one at each end close to the connector).

So it seems the monitor are just losing synchronization so they blackout until it recoers. Could be also related to some encryption (if you are playing some protected material, the signals may switch to so e encrypted mode, so it is supposed to become supposedly impossible to record).

Anyway, check the cables but also the monitors and the computer, if the shield connection is poor for any reason, it may become sensitive for disturbances...

Also are the computer and all the monitors plugged into  the same wall socket? E,g, into a common extension cord with multiple sockets on it.
The thing is, if the wiring, mainly the PE (ground) meet only at the distribution board, the disturbance between the grounds may become way too strong, even for a reasonably good signal cords.

For the sockets it does not matter that much where the lantern is plugged, but how close/far is the power wiring of the monitor vs computer merged together.
If the problem persist when using different connectors on the monitor/pc (not adaptors), the connectors won't be broken, ir is very unlikely for all to be broken the same way.
Because similar symptoms, I would guess the V or H sync pulses (even the digital format use them, to indicate next row/scree) are disturbed and the sequencer transferring data to the panel RAM (even the analog VGA does that, just via ADC's) get lost and so shuts down until it recovers.