I bought a home that was wired with a stereo system so we could access
music in each room.  Recently, after an electrical storm, it stopped
functioning. The brain is an Ambiance AM-16 Mainframe, and the system
includes an AX 816 multiplexer. I've seached for both and received no
matches. How can I repair it? I'm willing to pay a reasonable price.

Do either items have a company name on it?  Like you I am coming up
blank - but most things like that would have a company name and city
if not entire address that you could use as a starting point to track
them down.

I found "Ambiance Accoustics" but there wasn't anything there that
seemed like what you are interested in.

Ambiance is the name of the company on the item. I also found Ambiance
Acoustics thru Google, but it does not appear to be the same company.
I sent an email to this company but have not heard back. My system was
installed in 1991, and I believe the manufacturer, Ambiance, went out
of business.

Check this one out:

www.ambiancetechnologie.com/en

It's a canadian company based in Quebec, and their website is not
completely translated. The products you mention do not appear on their
catalog, but they do have very similar products. Maybe they have
discontinued the AM-16 and the AX-816, and they are no longer listed.
Try contacting them.
Hope it helps. Cheers! :)

Most of the time simple things get destroyed first, with a good
general knowledge of electronics and experience with a multimeter and
component (diodes, resistors, etc) installation/removal you should be
able to repair things yourself or anyone with such experience can.

However paying someone to do this can quickly add up fast at $50 per
hour, as although it is simple, it takes a long time to test
everything.

In general I'd say if the item is dated replace it, unless it would
cost a few thousand to replace.  Electronics are getting more and more
disposable everyday.