Option key doesn't present Start Up boot options on Mac mini

See: Install macOS on an external storage device and use it as a startup disk - Apple Support

You don't want to do this on a drive that has data on it you wish to keep. Make sure you back up anything on the external drive first.

brarn wrote:

But I think I have a problem with my ext drive in as much that I have 2.5 TB of data that I can't afford to lose nor do I have a way of backing it up.

Are you implying here that your external drive with important data is not already backed up? Is the copy of the data on the external drive the only copy of that critical data?

If the data on the external drive is your only copy & it is not backed up, then that should be your first priority if the data is important to you. Any important & unique data should be backed up frequently & regularly. The more important the data, then the more backup copies you should have of that important data.

Does your Mac mini have an Intel processor, or an Apple Silicon one?

Pressing the Option (⌥)  key to get into Startup Manager only works on Intel-based Macs. On Apple Silicon Macs, you are supposed to press and hold the power button until you see a startup options screen.

Mac startup key combinations - Apple Support

brarn wrote:

What do you think is the best way for me to try to salvage any lost data on my external HD?

The only way for you to salvage any data will be either to

• Pull it from backups, which you don't have; or
• Copy the data from the read-only (and likely corrupted) drive – which will involve going out and getting another drive, since you don't really have a place to put the data otherwise. This doesn't need to be a SSD - it could be a hard drive – but no place to copy the data = no recovery.

I suppose you could also take the drive to a professional data recovery service, if you don't mind spending maybe hundreds or thousands of dollars without any guarantee of success. They're going to want to copy the data from the drive with the problem to another drive, too.

Sorry, I was thinking about something else. You should have the keyboard plugged in as I mentioned, and, press the Option key before, or soon after you turn on power.

brarn wrote:

Yes, I want to change the start up disk from Mac hard drive to my external drive, because I don't appear to have enough space on the Mac HD to upgrade my OS. hope that makes sense...

That's fine. Do you actually have macOS installed on your external drive or is that what you are trying to do? The external drive won't show up when you hold the option key unless you actually installed macOS onto it.

Does it show up on the Desktop? If it does you can offload the necessary data to another drive to backup it up. Do you have Time Machine activated on the Mini? If so you should be able to backup the EHD as well.

Does the EHD show up in Disk Utility? If it does try running First Aid on it.

Assuming they are all in a folder, sure. just drag them to the external. Then delete the folder on the internal drive. I would mention how to "move" it rather than "copy" it, but it's safer to just copy, just in case something goes wrong.

The message says "You can still open or copy files on the disk, but you can't save changes to files on the disk. Back up the disk and reformat it as soon as you can."

Backing up that disk to another external disk sounds like a very good idea. You might want to back up to a new external disk – not one of your existing backups. (If the new backup fails in whole or in part – due to the issues that caused macOS to make this drive read-only – then you might want the option of being able to pull some of your data from your existing backups.)

As to reformatting that disk "as soon as possible" – there's probably no way around reformatting it to get it to a state where it is writable again. But unless you need that particular drive right away, the only really urgent thing would be copying as much data as you can off it. Once you've done that, you can take your time about "pulling the bandage off" and reformatting the corrupted/damaged drive. (Most likely, suffering from data corruption … not actually suffering from some physical hardware failure.)

What system are you running on the Mini and what system do you want on the EHD?

You have to hold the key down until you see, Loading startup options, or words to that effect. If you don't see that, try plugging in the keyboard with the charging cable first.