(Hat tip to xandram's Tear Apart the World idea for inspiration)
I've always been amazed by the idea of continental drift: that the Earth is constantly shifting imperceptibly beneath our feet and, indeed, has only been recognisable as the planet we know and love for the merest fraction of its history.
In the blink of a geological eye my country may be slammed into mainland Europe, the snow-peaked Channel Mountains rising thousands of feet above what was once Boulogne. Or, perhaps, we will drift gently out into the Atlantic to one day join the North American continent. Who knows?
 Of course this isn't possible since the UK is part of the Eurasian plate, but I like the imagery [/edit]
To offer people a better understanding of the history of the Earth, I propose a dynamic globe that can play out (our best guess of) a simulation of the geological history of our changing planet. Beginning with Rodinia, the supercontinent that formed around 1.3 billion years before present, the globe would display the gradual fragmentation and reformation of Pangaea, Laurasia and Gondwana until we reach the present day, along with as much information as we can recontruct about the rising and falling of the great mountain ranges of prehistory.
The simulation could play at whatever speed you like: over the course of a few seconds to teach kids in the classroom, or over the course of days or weeks to provide an amusing diversion when you're bored ("I'm just going to check on the world, dear, to see how the Pyrenees are getting on"). The globe would offer a relaxing form of entertainment, sort of like a lava lamp with a more clearly defined purpose and less association with blacklights and drugs.
To entertain while maintaining its educational value, the simulation could run in two modes. The first would remain faithful to our knowledge of continental drift, displaying nothing but reality. The second could run a rather more fanciful simulation, embellished with occasional volcanic eruptions and the like (and perhaps even displaying our best guess as to the early history of the planet as the surface cooled into a solid crust.
Off the top of my head, the dynamic globe could be implemented using three methods (to suit every budget):
1. Magnets and iron filings
A sort of lashed together Heath Robinson affair using a globe made up of multiple sections of magnetic material and covered in a layer of iron filings (which themselves are encased beneath a transparent layer of plastic to prevent them from getting everywhere when the magnets are switched off).
To generate the simulation of continental drift, sections of the globe are sequentially magnetised and demagnetised to attract the iron filings, allowing them to form continents and move smoothly across the surface of the globe. Unfortunately, the simulation run on this version of the globe would be limited to the display capabilities of little pieces of iron.
2. LCD screens [link]
The globe is covered in a number of small LCD screens, each playing a part of the sequence.
3. 3D holographic projection [link]
Expensive, difficult but geekily impressive.
Aside from recreating a simulation of historical continental drift, the globe could be programmed to play anything: a prediction of future continental movement, for instance, the effect of a large meteor impact, or environmental campaigners could use them to show the effect of rising sea levels in ever more harrowing and donation-inducing ways.
Naturally, the globe (or, at least, the version using LCD screens) would be capable of connecting to the Internet to download and display the latest meteorological data/updated geopolitical map data next time we decide to annex some third world hovel/topographical data to identify 'New Atlantis' when New Orleans finally vanishes beneath the waves.
In fact, any program could be loaded to the system - even on a religious theme: the earth remains dark for millions of years before suddenly springing to life, fully formed, 6000 years ago and remaining that way (with a brief interlude of flooding, during which all that can be seen is a tiny ark) until God becomes so annoyed with our sins that he crushes it all back into a fine dust with his massive, vengeful fists. This version, of course, would also include bonus footage of God burying all the fake dinosaur fossils to keep himself amused as he waits impatiently for the joys of Judgement Day.
There seems to be a widely held belief these days that we are destroying the earth with our nasty little ways: melting the ice caps, destroying the rainforests and so on. With the LCD screen option for my dynamic globe we could include a greater level of detail than simply the shifting continents: the expanding and receding ice caps, for instance, and the historical extent of rainforests (both of which we have a reasonably fair idea of). As well as offering an extra level of interest to the globe it may also help convince us of the fact that we are of little importance in the grand scheme of things: once the forests are all logged and the ice caps have melted and sent us to our watery graves the Earth will carry on just fine without us, as it did for billions of years before we bumbled along.