At the moment, it would tell us more about cosmology than about the underlying physics, were we not to detect the gravitational waves. We expect gravitational waves to be emitted from two black holes as they orbit very close to each other, just before they merge. From general relativity, we know how big of gravitational waves to expect. However, we don't know how many such black hole mergers occur in nearby space.

LIGO can only measure gravitational waves down to a certain strength, and so we can only detect waves that are caused by black hole mergers within some distance of Earth. Predictions of how many such mergers occur vary wildly, depending on the assumptions in the model. (I don't have exact numbers, and would appreciate if anyone does have the numbers.)

If we did not detect gravitational waves, I would use that as a lower bound of the number of black hole mergers, rather than as a breaking of general relativity.

Not observing any gravitational waves wouldn't prove that gravitational waves didn't exist, so there wouldn't be nearly as much fuss and news articles written. Imagine the press release: "Jury still out on existence of gravitational waves: LIGO not sensitive enough/ Is LIGO a gazillion dollar waste?." So while if someone were able to PROVE that gravitational waves didn't exist that would be pretty spectacular because it would upend decades of scientific research and cause us to rethink relativity.

But this is still pretty spectacular in it's own right. The predictions that physicists make are made by examining mathematical models that are only partially vetted. Some parts of the models remain theoretical because we don't have instruments sensitive enough to measure their predictions (we are talking about energies on a ridiculous tiny scale here). Finally we have invested millions on an instrument that is sensitive enough and it confirms (to the best of its ability) that we were on the right track. This is a great relief to the scientists that they don't have to go back to the drawing board and rethink all their models, plus it is a great victory for predictions made before the age of computers.

If they were not detected, that wouldn't necessarily say much unless you could demonstrate certainly that they also should have been. In which case, yes. Though, in actuality, the physics of gravitational waves have been indirectly observed previously [0], so it's not like it's still entirely in the realm of hypothesis.

[0]: The Hulse–Taylor binary pulsar: https://en.wikipedia.org/wiki/PSR_B1913%2B16

I actually was sort of curious as to the opposite. What does it mean if gravitational waves (that we can detect) are extremely common and become fairly easy to detect with future technology given todays news?

The article says many expected LIGO to take much longer but it found it rather earlier equating it to beginners luck (I know the article is written for the layperson but it leaves much to be desired with that phrase).

You never know we could have telescopes that use Gravitational waves instead of light.