rationalskepticism.org

It's the last remaining untested prediction of General Relativity, and it has some interesting consequences. Firstly, it can provide a window past our current observational limit of 380,000 years after the onset of expansion, because the universe was opaque to photons until that time. A workable gravitational wave observatory could potentially allow observation all the way back to the Planck time.

Also, dependent on the specific nature of the gravitational waves, some cosmologies can be falsified, such as the 'brane-worlds' model of Turok and Steinhardt.