Can lunar atmospheric density, as it exists today, increase naturally? Are there any mechanisms which could increase the density of the moon's atmosphere.

The moon's existing atmosphere is exceeding rare, with about 1 000 000 molecules of gas per cubic centimeter at the surface, ten trillion times less than the Earth. 

The source of this gas is the impact of solar wind on the moon's surface, and some outgassing from the interior of the moon. 

Can this density vary significantly? Could it reach a density comparable to Earth's? Are there natural processes which could cause the atmospheric density to increase, either in the short term, or over a longer time span.
There's not enough gassible (is gassible a word?   I'm guessing not), OK, there's not enough of the right elements to form enough gas on or in the moon to create any kind of atmosphere that would measurably slow down falling objects (say, enough to land a probe with a parachute, like they do on Mars).   Atmospheres are heavy.  They may not seem heavy but they are.   Mars' atmosphere has a mass of about 25 trillion metric tonnes and that's a very thin atmosphere compared to Earth.  

The Moon, with roughly 1/4th Mars' surface area, would need about 1/4 of that mass in it's atmosphere to have an atmosphere with similar air resistance to Mars, somewhere in the 6-7 trillion metric tonne range.

There's some ice on the moon in craters that never get direct sunlight, but not nearly enough ice to make that kind of an atmosphere.   The Moon also has very infrequent volcanic eruptions, but while some gas is likely released, again it wouldn't be nearly enough.

One possible scenario would be a large asteroid or comet impact that could both release some trapped gas from below the surface and release gas from the impactor.   The problem with such a scenario is that the heat from the impact would likely send much of the gas away from the planet at above escape velocity shortly after impact.    The escape velocity of the Moon is 2.38 km/s and at the top of this theoretical atmosphere it would be a little bit less than that.  A large impact from a comet is the best way I can think of, but that wouldn't be very efficient.

Even if you found a way to create that kind of atmosphere on the Moon, it wouldn't last very long.  The Moon is simply too small, with too little gravity and it's too close to the sun to maintain an atmosphere, unless that atmosphere was Xenon or if you want to be a bit more daring, sulfur-dioxide (acidic when it comes in contact with water and stinky) or sulfur-hexaflouride, which isn't toxic in and of itself but becomes very toxic when exposed to UV light from the sun, so, lets not use either of those, but Xenon is safe, so we can use that.

see atmospheric escape velocity.   



So the answer to your question is yes, the Moon could be given a thick enough atmosphere to slow falling objects down, but it would probably need to be done artificially, as there's no natural process I can see that would do that, and ideally, you'd use a very heavy and non toxic gas like Xenon.    All the Xenon in the Earth's atmosphere wouldn't be enough.   A rough calculation, Earth's atmospheric Xenon comes to less than a billion metric tons, which is still several thousand times too little mass to create a mars like atmosphere on the Moon.   Now if you harvested all the Xenon from Jupiter and/or Saturn and moved that to the moon, then you might be able to create a somewhat long lasting atmosphere on the moon that would provide some measurable air resistance.   It wouldn't be breathable, but it wouldn't be toxic, which is a plus and with enough Xenon, you might be able to get around on the moon with just an oxygen tank and no pressure suit.  Buildings with Oxygen/Nitrogen would need to be air-tight but not vacuum resistant, which is far easier to build.   Enough Xenon would make living on the moon quite a bit easier.   Still wouldn't be easy, but definitely easier.