Is there evidence for water on the moon?
Yup! Quite a bit actually, all in the form of ice.
Common sense would tell you that there's no way that water could survive on the moon given its exposure to the sun - you'd think that it would all just photodissociate( separate into oxygen and hydrogen) due to solar radiation. However, there are numerous craters on the moon that stay in constant shade and are thus extremely cold - this is where you could find ice near the surface.
In 2009, remote sensing data showed absorption features in the 2.8-3.0 µm region, which is indicative of either hydroxyl or water bearing minerals in silicates. So water-bearing minerals have essentially been directly observed, and it's posited that there might even be large ice sheets at depth in the moon.