Try cycling the breaker off and on, once, to see if that silences it.

Well, Randy, you're a genius! Turning the breaker off and on seems to have eliminated the buzzing! Does that mean the "problem" is solved or does it tell you that we have a serious problem that needs to be fixed?

Thanks,
Worry-Wort

If turning the breaker off still de-energizes the circuit, as I believe it will, problem solved.
One exception - if it's an old Stab-Lok panel and breaker, they are known to be unreliable and the cause of some house fires.

Uh-oh. Our panel box is a Stab-Lok. YIKES!

From your post, I'm guessing your ultimate recommendation would be to replace the panel box. But, is there something we should do right now? (Right after we make sure our fire insurance is up-to-date?! Aghh!!!)

There are still many Federal Pacific Stab-Loks that haven't been a problem, and I don't believe there has been a recall, but their pre-1990 equipment has had a number of failures with 'dramatic' results. A Google search will turn up a number of hits on the topic.

I'd make sure that, for each breaker, turning it off does indeed deenergize that circuit. Taking the cover off of the panel, if you are comfortable doing so, look for visible signs of something 'wrong' that indicates excessive heat or deterioration, i.e melted insulation or plastic.
I'd then start getting bids and budgeting for an electrician to replace the panel. JMHO

Randy, thank you so much for your help! Thankfully, we don't see any signs of melting or excessive heat. Will start making phone calls to electricians soon! Thanks, again!

The very old stuff is usually fine.

Quality declined and problems started occurring.

i do not remember the associated dates, and the parts have gotten expensive enough to replace the panels proactively.

Most new panels will use less expensive breakers tan trying to find a StabLock anymore.

A lot of the problems started as two pole breakers failing to trip mutually (as required).