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EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no longer relevant".

As some of you may already know, there are plans in the making for a Museum of Mathematics in New York City. Some of you may have already seen the Math Midway, a preview of the coming attractions at MoMath.

I've been involved in a small way, having an account at the Math Factory where I have made some suggestions for exhibits. It occurred to me that it would be a good idea to solicit exhibit ideas from a wider community of mathematicians.

What would you like to see at MoMath?

There are already a lot of suggestions at the above Math Factory site; however, you need an account to view the details. But never mind that; you should not hesitate to suggest something here even if you suspect that it has already been suggested by someone at the Math Factory, because part of the value of MO is that the voting system allows us to estimate the level of enthusiasm for various ideas.

Let me also mention that exhibit ideas showing the connections between mathematics and other fields are particularly welcome, particularly if the connection is not well-known or obvious.

A couple of the answers are announcements which may be better seen if they are included in the question.

Maria Droujkova: We are going to host an open online event with Cindy Lawrence, one of the organizers of MoMath, in the Math Future series. On January 12th 2011, at 9:30pm ET, follow this link to join the live session using Elluminate.

George Hart: ...we at MoMath are looking for all kinds of input. If you’re at the Joint Math Meetings this week, come to our booth in the exhibit hall to meet us, learn more, and give us your ideas.

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    $\begingroup$ I'm reminded of the following quote, which perhaps would be good to include in the museum: "Numbers exist only in our minds. There is no physical entity that is the number 1. If there were, 1 would be in a place of honor in some great museum of science, and past it would file a steady stream of mathematicians gazing at 1 in wonder and awe." - Linear Algebra by Fraleigh + Beauregard $\endgroup$
    – Zev Chonoles
    Commented Dec 25, 2010 at 20:38
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    $\begingroup$ What an opportunity! Clearly, the fact that many of us mathematicians ourselves don't even know about this project (or related ones mentioned in other responses) means, above all, we need to hire marketing professionals! And designers should build the exhibits. (But as for content, I've always liked the Borromean rings: en.wikipedia.org/wiki/Borromean_rings) $\endgroup$
    – Eric Zaslow
    Commented Dec 26, 2010 at 3:09
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    $\begingroup$ I'm wary of both marketing professionals and designers. We are interested neither in selling junk people do not really need, nor in trying to beautify something that is ugly by its nature. If anything, we should get a few high level math. people with good taste and some knowledge of the outside world to make decisions about what to do. But I doubt it'll be done. I bet Percy Diaconis, say, has been neither invited as a consultant, nor even told of the project. $\endgroup$
    – fedja
    Commented Dec 26, 2010 at 15:34
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    $\begingroup$ @fedja : While Diaconis doesn't seem to be involved, the advisory board (listed here : momath.org/about/advisory-council) includes a lot of very good mathematicians, for example Bjorn Poonen. That being said, I'm still pretty skeptical that a "museum of mathematics" is possible... $\endgroup$
    – Andy Putman
    Commented Dec 26, 2010 at 20:34
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    $\begingroup$ @Timothy : My skepticism comes precisely from the math sections of a number of science museums I have been too. They've all been pretty lame (and that's not just Andy the math-snob talking -- my wife and kids haven't enjoyed them either). We just don't have cool things like robots or spaceships or dinosaur bones or life-size models of the human heart to show off! $\endgroup$
    – Andy Putman
    Commented Dec 26, 2010 at 23:18

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I'd suggest an interactive exhibit where people can tweak the parameters of a population model with 3 species in it. Have an information panel which explains what the parameters represent. Suggest goals such as (1) keep the rabbits from going extinct, (2) create a stable equilibrium where all three species survive, (3) find a cyclic solution, etc. Experts could probably come up with a good system that exhibited lots of interesting behavior. Let people visualize their solutions both graphically (3-D graph of all three populations as well as 2-D graphs of any two populations of their choosing). There are probably other clever graphical representations that others could come up with as well.

Overall, I think this is a great endeavor and I hope that you will focus on making the exhibits interactive, with pathways to learning. Ideally, the same person could visit an exhibit a half dozen times and learn something new each time. Of course, making things fun and interesting is really important too - but I think that should naturally emerge from the design of interactive ways to explore a beautiful piece of mathematics.

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The Body Counter: Making Math Matter in Human Rights

An exhibit explaining the use of multiple systems estimation (MSE, a type of statistical inference) in human rights investigations and trials, based on Tina Rosenberg's article The Body Counter.

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An original S.I. metre bar.

Who wouldn't love to see one of those in person?

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    $\begingroup$ I wouldn't. Talk about non-canonical choices. $\endgroup$
    – Qiaochu Yuan
    Commented Dec 26, 2010 at 20:59
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    $\begingroup$ The SI metre bars are pieces of math and science history. I think it could be part of a great exhibit. $\endgroup$
    – Ben Warren
    Commented Dec 27, 2010 at 3:24
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    $\begingroup$ Actually, I would like to see that. One can see a meter stick anywhere, but seeing an original bar would concretize the notion that a great deal of thought went into the meter. It's at the BIPM. I'm not sure how willing they might be to let it go or loan it: en.wikipedia.org/wiki/Metre $\endgroup$
    – Anthony Pulido
    Commented Dec 27, 2010 at 8:03
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    $\begingroup$ I don't understand how it's math history. It seems almost like engineering history. $\endgroup$
    – Daniel Moskovich
    Commented Dec 28, 2010 at 21:24
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  1. Models of the sphere eversion that emulate the chicken wire ones. I think someone made some for Morin at one time.

  2. Interactive computer graphics of Seirpinski $n$-simplices, interactive computer graphics of hypercubes, the 120 cell, and the 24 cell.

  3. Transparent models of knotted surfaces.

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    $\begingroup$ Hypercubical arrays of Hasse diagrams for divisibilities. $\endgroup$
    – Scott Carter
    Commented Oct 27, 2011 at 3:32
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Something like the clickable math atlas: http://www.math.niu.edu/~rusin/known-math/index/mathmap.html

but the clicks should probably lead to places showing for laypeople what these different fields are!

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Nothing. In a museum you put thing that are obsolete now. In mathematics nothing is obsolete (yet).

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    $\begingroup$ Actually, putting the empty set in the museum, would not be a bad idea at all. $\endgroup$
    – Lucas K.
    Commented Dec 26, 2010 at 16:48
  • $\begingroup$ So painting it dead then ? Movies too and music certainly! $\endgroup$
    – Jérôme JEAN-CHARLES
    Commented Jan 9, 2011 at 0:14
  • $\begingroup$ You're thinking of a house of mathoms. A museum needn't always be such. $\endgroup$
    – Zsbán Ambrus
    Commented Jan 28, 2011 at 10:17
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