Just curious:
What/where in St. Louis is the place where this would be? I remember visiting a really fine children's science museum there about 25 years ago.
Obviously, there are many ways to accomplish what you are asking about, beginning with cascaded decade counters and going up all the way to--gasp!
--a microprocessor.
I think optical beam interruption (visible light or infrared) may be a better choice than a touch sensor.
You might consider a little bit of delay time (a very few seconds) before the sensor (whichever you use) is able to be triggered again.
There will always be people who will touch the thing repeatedly just to watch the numbers change. Since your sensor can't tell people apart, perhaps the sign should read "This meteorite has been touched [numeric display] times since [date installed]!"