Did we ever witness a "meteor shower" which was actually dust or debris of some kind from our own Moon?
There is a class of meteorites called Lunar Meteorites and several of those have been found on earth.
Of course, it doesn't necessarily indicate whether we have actually witnessed a certain lunar meteor shower. Considering that nearly 100 metric tons of space debris falls on earth each day, one would imagine that it'll be colossally impractical to attempt mapping a meteorite to the moon, using a reference to a recorded meteor shower.
However, it might be worthwhile to take a look at this link which addresses rather interesting questions:
How did lunar meteorites get here?
Any rock on the lunar surface that is accelerated by the impact of a meteoroid to lunar escape velocity of $2.38 km / s$ or greater will leave the Moon's gravitational influence. Most rocks ejected from the Moon become captured by the gravitational field of either the Earth or the Sun and go into orbit around these bodies. Over a period of a few years to tens of thousands of years, those orbiting the Earth eventually fall to Earth, whereas those orbiting the Sun may also eventually strike the Earth in a few tens of millions of years after they were launched from the Moon.
How do we know that they come from the Moon?
Chemical compositions, isotope ratios, mineralogy, and textures of the lunar meteorites are all similar to those of samples collected on the Moon during the Apollo missions. These meteorites are rich in a mineral called anorthite and have high concentrations of aluminum and calcium. Because of some unique aspects about how the Moon formed, the lunar highlands are composed predominantly of anorthite. To the best of our knowledge, anorthite is much less common on asteroids, and on the surfaces of any other planets or planetary satellites.
Quotes derived and paraphrased from the linked article.