What is "other"??

D!rk Said:

"What is "other"??:[...]"

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Q1. "I don't understand how the "other" section could grow from around 50 GB to over 75 GB in a day. Was that related to my abortive attempt to install OS Big Sur, maybe?"

A1. It might be, and it would seem to be the case, being that you were making such a large change to your Mac: a macOS installation.

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Q2. "Although it claims to need 35 GB for installation, that apparently wasn't enough, and it stalled midway and said insufficient space. System files are 11.33 GB, which sounds about right."

A2. System file are fine, numerically. IT likely stalled due to little space available.

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Q3. "Why can't I even get a look at what is in the "other" category? The reinstall sounds like major surgery, even with a backup."

A3. These are miscellaneous file found through the install. The reinstall is the ultimate way around this. If it is too technical for you to proceed with getting this serviced:

To Get this Mac Serviced...

Contact Apple, and setup a Genius Bar Appointment or see an AASP (Apple Authorized Service Provider) in your local area.

Contact Apple:

• Contact Us - Choose Locations
• USA: 1(800)MY-APPLE
• Proceed from there as Necessary

Or...

Contact an AASP: 

1. Go Here: Find Locations - Apple Authorized Reseller
2. Click: Service & Support
3. Enter: your location information
4. See: if there is an AASP nearby
5. Contact: an AASP that shows up, and find out more about the services that they offer to fix the iPhone

D!rk Said:

"What is "other"??: Of my 121 GB of space on the hard drive of my Mac Mini, over 71 GB is listed as "Other".[...]Also, is there a tool to defragment the hard drive, in order to make it store more efficiently and smoothly - or is that a PC thing only? Which files can I move to my 1 TB external hard drive, and is there a downside to doing that?"

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Clearing "Other":

"Other" basically contains unnecessary files such as caches that are used for system ease of use - all of which gather over time: 

Reinstall the macOS:

Nothing would be deleted. It's just that items that have become corrupted or deleted, would be replaced or reinstalled back to where they should be. Also, all of this "Other" is intended to be deleted after the reinstall. Go Here: How to reinstall macOS from macOS Recovery - Apple Support. Important: Be certain to create a backup of your Mac prior to all of this. That way, you can have something to restore from the Mac from, should anything go wrong with the reinstall.

In Apple's See used and available storage space on your Mac, there is an excerpt that states that there is nothing to be done by you as the operating system manages the Other category: