Remove Water Mark from Eames Walnut Stool  

The
THe first thing I would try is some fine grade wire wool with some oil.
others may have other ideas ??

Careful with the steel wool
I've found that the metal fibers can leave dark marks in the wood; would suggest very very very fine sandpaper instead.

wood bleach
The best way to get a dark water mark off of wood is to remove the finish with a finish remover or light sanding. Then you need to bleach the wood using wood bleach found at your local hardware store. You will then need to re-oil the wood with tinted danish oil or re-stain it and finish it whatever finish you want to protect it.
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It probably won't work but won't do any harm, try getting some silica gel beads (old shoeboxes...) and make a little sealed envelope of them over the top of watermark, leave it for a couple of days.

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. . .then wave a red silk handkerchief over the envelope, mutter the words "RayandCharles, RayandCharles" any odd (not even) number of times, and (finally) remove all your clothes and go to bed.
In the morning, everything will look exactly as it did three days earlier !
Seriously, that's probably too delicate an attempt at remedy. (Even if the stain were still wet, the silica gel would not be likely to remove the swale at the perimeter of the stain, I think. And why a "sealed" envelope ? )
But I can't say what is an effective thing to do. Scotchbrite is a non-metallic substitute for steel wool, however, and is made in several "grits."

Steel Wool
Use only 0000 steel wool and buff with the grain. I use this with a liquid furniture polish to put a final buff on newly varnished pieces. Buff gently WITH THE GRAIN!. Do NOT buff dry. Also try cigarette ash rubbed in with a damp cotton cloth. Here's a set of 50's nesting tables I refinished this way.

I did say it probably wouldn'...
I did say it probably wouldn't work but silica is used as as a dessicant in the timber industry.

Yes.
But once water has stained something, we're past the problem of removing the moisture, aren't we. Sorry, I wouldn't take away from your helpful effort. And I don't know what the best remedy is, besides stripping, sanding, and refinishing. And it's possible that bleaching and recoloring would be necessary, as suggested.
What exactly is a water stain is a good question, maybe. How does colorless water permanently darken wood under finish ? Or does it ?
The swale I mentioned (I got the term from a dry cleaner friend) is the place around the perimeter of a stain, marking the place where a spreading stain stopped spreading. I envision whatever soil is carried by the staining agent being left there, like a bathtub ring.

I've heard good things about a product called
the "Jasco Ring Remover Cloth". I assume it works on all water marks-- ring shaped or otherwise.
Other heresay cures:
mayonnaise applied for a half an hour, then wiped off.
warm iron (no steam), using a thin cotton cloth between iron & wood.
(disclaimer: you may find yourself in need of the Jasco Mayonnaise Remover Cloth, or the Jasco Scorch Remover Cloth, after attempting either of these cures.)

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Hi SDR, I imagine either the water brings a litle tannin closer to the surface or there is a little oxidation, suede marks like this too.
I would be tempted to chuck the whole thing back on the lathe 🙂

For sanding fine woods, the...
For sanding fine woods, the black sanding paper used in the car industry is often the best. It's very very fine grain isn't as harsh as regular sanding paper and doesn't scratch the wood's surface...

"White rings, caused when...
"White rings, caused when water vapor penetrates into a finish, can be removed by wiping them gently with a cloth barely dampened with denatured alcohol..." (this old house) http://www.cnn.com/2008/LIVING/homestyle/02/05/dinged.furniture/index.html
or use your search engine, google is nice. type 'removing water rings from furniture'.
it is worth trying some of the gentle hints first.
often, with an older finish, refinishing the entire surface is necessary.
http://www.cnn.com/2008/LIVING/homestyle/02/05/dinged.furniture/index.html

I'm a bit nervous
Naturally I don't want to do any of the harsher remedies just yet. I did do a google search and found alot on Mayonnaise mixed with ashes and a blow dryer?? I tried just the blow dryer got nervous after a minute and turned it off. The last thing I want to do is more damage!

The various cures undoubtedly apply to different wood-finishes
Perhaps you should contact Herman Miller to determine what kind of finish is on the stool, before proceeding.