Can't add a certain person's name to his photo

You wrote:

"A list of all the Andys in my address book appears, and I click on his name. Nothing happens. The list goes away, but Andy is still named 'unnamed'."

This can happen, if Andy's photos have been merged with someone else. To fix this, try to create a new "People+Pets" album for Andy, without using his Contact card. Enter a slightly different nickname where it is showing "unnamed", then hit "Return" and ignore the suggested Andys.

Can you create a new Andy album this way? If you can, check all People albums for other people, if there are some misnamed faces of Andy and reject the wrong name. Then assign the new name for Andy.

You can switch back to the original name from the Contacts, once the new album for Andy is working again.

Try to

Find a picture of Andy that has a name, and change the name to Andrew. If that works, then there will be a new People album named Andrew.

The trick, as léonie was saying, is that Photos may have learned that Andy's features have been conflated with someone else. If you find Andy's name attached to someone else, you may have to rename that person, as well, and produce another new People album, there, too. Changing the names makes the old albums empty so that Photos can unlearn those attributes. léonie is the expert on this.

Let us know how it goes…

Martin S Taylor wrote: I can't find a picture of Andy that has a name. All of the pictures of him are "unnamed".

And you can't add any name to the "unnamed" Andys? Wow-- that's weird! Usually, "unnamed" means that photos recognizes that this as a face. That's the hard part, and from there it's just regular computer information database stuff.

Just to narrow things down, try these steps, one at a time:

• Restart the computer (of course)
• Re-start in Safe Mode. This bypasses certain potentially disruptive processes, and it often helps. Safe Mode is different for different computers, so see this: 

           Start up your Mac in safe mode - Apple Support 

• Use another library if you have one, or make a small test Library with a few pictures in it and see if the same thing happens. You can create a new Library by closing Photos and then option-clicking on the Photos icon.
• Make a new user and see if the same problem recurs with it.  This is a bit harder, because a new user can't access a different user's stuff. You would need to move your Library out of your own user's folder up to the general "Users" folder. 
• Rebuild your Photos Library--close Photos and re-launch by option-command-clicking the app icon. If you use iCloud Photos, it’s possible that this will cause a re-sync that lasts long enough to make you nervous.

The idea is to find out if the problem is with the Library, with your account, or with caches and login items, and stuff like that. One of these steps may fix the problem, but their failure to work also helps figure out what's happening.

Let us know what happens…

Give this a try:

1 - create a new, empty test library.

2 - import a couple of photos of Andy

3 - try adding his name to the photo.

4 - if successful close the test library, open your current library and try adding his name again.

5 - if the test library is unsuccessful log into another user library and repeat the test library there. That will tell us if the problem is system wide or limited to your user account.

Martin S Taylor wrote: repairing the library…seems to have done the trick.

Great! As a test I had made a new Library with 1500 pictures, all with faces, and over night it had recognized all the faces that it had seen in the Library that they had come from. Being asleep, I don't know when it really finished, of course, but I thought that was pretty good. It shouldn't take too long.