Moonbow?

What is the name of a rainbow produced by moonlight? Does it have a different name?

Yes, it’s called a moonbow.

There are other optical phenomena associated with the moon. The most common is a halo around the moon which is caused by ice crystals way up high. Moonbows are relatively rare in comparison.

I’ve never seen it myself, but Cumberland Falls in Kentucky is well known for producing moonbows during full moons.