Hypothetically speaking, what might happen to Earth if one day the moon became un-tidal locked and started to rotate?
Not much would happen to the Earth, but the lunar interior would heat up a bit.
The backstory: Both the Moon and the Earth exert tidal forces on each other, deforming their shapes. But the deformation has a time lag, so it is not perfectly lined up along the Earth-Moon line.
Long ago, the Moon rotated faster than its orbital period, and it was closer to the Earth. Gravity acting on the time-lagged tidal deformations enables the two bodies to share angular momentum. The Earth and the Moon both rotated slower and slower, and the Moon moved outward into longer and longer orbital periods. Eventually the Moon became tidally locked, and as it continues to move outward, it will continue to rotate slower and slower, maintaining the spin-orbit resonance.
Your scenario: The Moon's new spin period would create a time-lagged tidal bulge, and the Earth's gravity would pull on that bulge until the spin-orbit resonance was restored. But the tidal forces in the Moon's interior would not be uniformly distributed, creating some friction and producing some heat. I don't have a good intuition about the amount of heat, and it would of course depend on the Moon's new rotation period.
The diagram above from Tufts is meant to explain why the Earth's rotation gradually slows, but the exact same forces would act on a Moon whose spin period is shorter than its orbital period.
While the side effects of whatever caused the moon to start rotating could be dramatic, there would be little direct consequence of the moon's rotation.
Newtonian gravity acts on particles and the rotation of a spherically symmetric body has no effect on its gravitational field. So the moon would stay in the same place, tides would happen as normal and the phases of the moon would be unaffected. Relativistic effects such as frame-dragging would be too small to detect.
Over time, the moon would be recaptured into a tidally locked rotation. This would take relatively little time, on a geological scale.
Of course, to change the moon's rotation by a significant amount would require a massive amount of energy. There's not much that could quickly get the moon to change rotation that would not have pretty serious side effects. on the moon and on the Earth too, but these would not be a direct effect of the moon rotating.
And as commenters have pointed out, the moon already does rotate, once per month, but I think we know what you mean: rotate faster (or slower) so that tidal locking is broken