My mother has a designer coffee table which was very expensive forty
years ago and is probably a museum piece.  I'd like to find an
appropriate place to get its designer identified and get it valued.

use your local yellow pages and look under appraisers 
yp.bellsouth.net

Oh, come on, if it had been that easy, we wouldn't need help.  We're
out in the sticks--there are no available appraisers.

Your best bet at this point is to take some quality photos of the
peice from avrious angles (and underneath) and put them on line for a
potential researcher to look at. Take photos of the top, the bottom,
the legs and the feet. If there are any drawers we'll need to see the
inside of the drawer, the bottom of the drawer (pulled out), and any
markings and any hardware on the peice.

Regards;
tutuzdad-ga

Sorry, that should have read: "...from various angles..."

LOL , I am out in the sticks too and recently had some antique
furniture appraised for insurance purposes , th appraiser i found was
more then willing to drive out here and do his work (afterall he does
get paid for it) and was actually drooling over some of the pieces ,
If you happen to live in Georgia I am more then willing to give you
his contact information