You can read for yourself why this material is such a hot area of research currently, but pay attention to the end of the article.
Geim and his co-workers suggest one such. Normal electrons can tunnel through potential barriers: as the barriers get higher the tunnelling becomes exponentially less likely to happen. Yet theory suggests that at very high speeds and with very high barriers, electrons will start to find it easier to get across as the barriers get taller. This is called the Klein Paradox, and demonstrating it in the universe outside graphene would need the sort of conditions you find close to an exotically heavy nucleus or, if you're particularly keen, a black hole. In graphene, you may be able to recreate an appropriate barrier with an ordinary semiconductor junction: there's talk of desktop super-colliders.
Other areas that graphene may illuminate include direct observation for the first time of relativistic electrons generating their own anti-particles – and there are even suggestions that some aspects of string theory, long thought to lie well beyond practical observation, may become testable.
And this shouldn't surprise anyone, especially if you're read my take on the physics that we learned from condensed matter. Many condensed matter systems can become the "testbed" of some of the most complex and exotic theories of nature.