After adding a menu entry in the grub menu due to adding a second OS, the arrow keys no longer work, making operating this menu very difficult. The OS'es in question are Linux Mint and OpenBSD. Where should I start troubleshooting, and what can be done about it?

Step #1 would be to restore the changed files from your backup so grub acts properly. Verify proper operations.
Step #2 would be to make ONE simple change again, and see how that changes the behavior.
Step #3+ repeat #2 until you get to the change that breaks things - back that one out and research your work: there is something wrong with that step.

additionally you may want to explain exactly what did you change.

All files have been restored to their original state, except for one added file that adds the alternate OS. Despite this, the behavior of the menu remains the same, none of the arrow keys work.

Maybe you should name that added file and show its content.

restore the original content and check if that works. Add your alternate OS. Explain what did you modify, how did you do that.

Move the file you created in the /etc/grub.d directory which contains the entry to chainload OpenBSD. Run grub-mkconfig and reboot to test. If the arrow keys work properly after this change, then you know there is a problem with that file and if you post it here, someone might be able to point out the problem. Out of curiosity, do your arrow keys function properly after you boot whatever you can boot?

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If that file is REALLY the only change, then that file may be the problem. Move or hide it and try again.

Content of /boot/grub/custom.cfg:
menuentry "OpenBSD" {
set root=(hd1,4)
chainloader +1
}

I first had the file in question in /etc/grub.d, with the result that OpenBSD was not added. The grub-mkconfig command doesn't make a difference. The arrow keys work well in both Linux mint and OpenBSD. The keyboard is a new one, so that can't be the problem.

No idea what the keyboard arrow problem is but have you tried just putting the OpenBSD chainload entry in the /boot/grub/grub.cfg file and saving the change. Do not run grub-mkconfig but just reboot to test. Mint seems to have some different file in the /etc/grub.d directory from your other post.

Did you ever try just putting the BSD chainload entry in the 40_custom file in /etc/grub.d and updating grub?

This also yielded no results. Maybe if I use grub-mkconfig it will work, but I haven't tested that.

That was already tested in my previous post, and it yielded nothing.

If you create a custom.cfg file that loads the BSD kernel directly instead of chainloading, are the arrows still inoperable? Does a stab or two or three of NUM produce any difference?

This is pointless. I mean stating something doesn't work is not really heplful it does not contain any information about the details.

That was already tested in my previous post, and it yielded nothing.[/QUOTE]

Was requested already to tell us exactly what did you make. Telling us "tested but did not work" means nothing. Probably you made a mistake somewhere which cannot be seen. Probably you missed something which cannot be detected.
So return back to the original working setup and explain line by line what did you make, what changed. Without details you will need to catch that missing step.

No because when you run 'update-grub' on one of the Ubuntu's (like Mint) you are running grub-mkconfig because the /usr/sbin/update-grub file is nothing but a stub/link to the grub-mkconfig script which is what actually runs. You can see this by simply opening the file in a text editor.

I agree you are being to cryptic in your responses to get help, you need to post more information than 'no results'.