Set up SSH on it and make sure you're able to connect into it while it works normally, that way when the issue occurs you can do a quick test to see if the system itself is still up and running.

I'm not on Linux Mint / Cinnamon but I've occasionally seen GNOME sort of hang/freeze so the screen display becomes non-responsive. After a couple of times of that happening I ended up setting up SSH on the system and configured a SSH client on my phone so I can do a SSH connection into the desktop and force-logout my user (which apparently fixes the issue and brings my main desktop back to a normal login screen). I haven't quite figured out if it's Gnome issues or something to do with my Nvidia GPU.. though with Linux if Nvidia is involved then it's usually Nvidia, ugh.

Also if you're physically at the computer when it happens try unplugging/replugging in the monitor cable, maybe there's something wonky going on there or with the display connection.

Just some ideas to help you along :)