Have you attempted to check if your OS is actually running when nothing is being displayed on monitor? Assuming it's Windows does it respond to shutdown in a proper manner when attempted to shutdown (assuming you know the keyboard sequence) confirming that indeed OS is running fine but there's no display.
Whenever it works fine, try to lower the resolution of your display thru OS settings and run for a few days on lower resolution and check if the issue goes away.
What if you just run the system with very minimal components outside of the cabinet, no HDDs connected. Just mobo ram and keyboard?
Do you have on board graphics or discrete graphics card?
Whenever display works, access BIOS and try altering the Video Buffer memory size in your BIOS
Try connecting the monitor to another power socket. Is it on separate mains socket or all are part of the same spike guard?
As already stated above, check if the motherboard has excess dust, try to clean it carefully with a light brush or if you have/can manage IPA cleaner.
Do you have a speaker connected to mobo? Do you hear a single beep when it works fine and other times when display does not work, no beep at all?
If you have just one stick of RAM try alternating it between available DIMM sockets.
As already stated above, if you have only onboard graphics, try if you can manage to get a suitable graphics card from your friend or thru a local repair shop and test with it as well.
Often it crashes in the middle of something
What crashes? Is it the OS or display suddenly goes blank?
Do you or your friend has a laptop where you can try hooking the same monitor in question and try to send the laptop display on that monitor? Does it work correctly then?