Well, on a hunch, I just solved my problem as I was preparing to ask it. Will share in case it's helpful to someone.
GOAL: Rename a mailbox to a time period and archive.
PROBLEM: Saved emails missing after renaming a mailbox, AND the old mailbox name reverted.
Tried Time Machine with Mail active - no dice. The monitor screen flickered, but no time line.
THE HUNCH: My email is aol, so went to the online site to see if the mailbox with either name was there. The new box name was there, with all the emails. BUT Mail still had the faulty mailbox & no emails.
SOLUTION: Quit Apple Mail, then re-opened, thinking it would pick up the changes to correspond to the server. Hurray! - it did!
LESSON LEARNED: The tried and true, "Quit an app that is not behaving, and re-open to see if it is fixed."
P.S.: Might have tried "get mail" from Mail, but didn't think of it.
"For what is age but youth's full bloom,
A riper, more transcendent youth" - Oliver Wendell Holmes