Do I need a relay to control a monitor's power with an Arduino?
It's always better to use MOSFET as a switch (or relay) like this:



It can handle loads up to 24V, 5A. The load is powered from an external supply. It is based on a IRF520 Mosfet and cost 0.55 a piece in Aliexpress. Search "Arduino IRF520 module"

You need one Arduino pin to switch on/off the load.

It's very handy and provide posts for power input/output.

I understand that you will still power the monitor from the wall, replacing only the original switch with Arduino. Otherwise, you will need a bigger relay