I used to have a 5.25 bay gadget years ago that was a fan controller/temp sensor with a display and clock etc. installed. It showed the time even when the PC was off, and it was made by Silverstone, if I recall correctly. TBH I never really liked it - the fan controller element was never brilliant (not as good as the PWM controlled motherboard based controllers), but most annoyingly it ran the clock off a single CR2032 coin type lithium battery, and it literally needed changing every month, sometimes more often if using a cheap battery. Failing to change the battery would cause it to lose time each time it was powered down, and annoying beeping alarms to go off randomly.
I don't know if they still make them, but there's something along the same lines here - https://www.amazon.co.uk/Socobeta-Automatic-Detection-Sensitivity-Controller/dp/B08RSR736M/
Honestly, I would suggest trying to dissuade your wife that they're a good idea. Offer to buy a standalone digital clock instead - there are many stylish options! If she insists, then I'd consider using double sided tape to stick one on to the front of the computer...