If the moon became a black hole, how much would it eat in a day?

It'd suck in very little mass.

One thing to understand about black holes is that they have super-strong gravity, but only when you are very close to their event horizon. Otherwise they're just normal objects. If the Moon became a black hole, it would have a radius of about 0.1 millimeters. You need to get pretty close to this distance before you notice anything special, and even closer to get sucked in. Further out (especially beyond the current Moon radius of 1737 km) it's completely normal.

Since space is mostly a vacuum and you have to get so close (compare that size of a millimeter to the distance from the Earth to the Moon, which is about 380,000 kilometers) to be sucked in, the Moon would not eat much a day.

It would suck in basically nothing. If the mass of the moon were concentrated in a black hole, you could draw a sphere around that black hole the size of the moon, and for everything outside that sphere, nothing would have changed. The gravity field would be just the same as it is now.

Now, black holes are tiny. At the mass of the moon, the radius is on the order of $frac{1}{10}mm$. The effects of the strong gravity gradient will be significant within a larger radius, but anything that is supposed to end up within the black hole must eventually hit this $frac{1}{10}mm$ sphere. Matter that's on direct collision course will hit, of course. And some matter that passes really close and looses enough momentum due to Bremsstrahlung (gravitational or electro-magnetical) or collision with other matter will become bound to the black hole, circle it while radiating away its energy, until it comes close enough to get sucked in. But the radius where this effect works is also rather small. Most matter that passes by simply gets deflected a bit towards the black hole, and that's it.

Secondly, if you hit the black hole with a stone, and it catches some stuff for feeding, the fall into the black hole will release enormous amounts of energy. This energy will heat up the stuff that's circling the black hole, and it will blow other matter away. So, even if you hit your black hole with a slow stone, almost nothing of that stone will actually end up being sucked into the black hole. Most of it will explode with the force of a nuke.

Thirdly, tiny black holes radiate energy via Hawking Radiation. They are not black at all. I don't know the exact amount of radiation that a black hole with the weight of our moon would emit. That's more radiation that tends to heat up matter before it gets a chance at being sucked in. The Hawking Radiation of a black hole with the mass of the moon is negligible and has no effect on its ability to accrete matter. G. Smith has thankfully provide the link to an answer on this site that lists the radiation power of tiny black holes against their respective mass.

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Bottom line: A moon-sized black hole won't suck in any appreciable amounts of matter. Nevertheless, random encounters with asteroids may happen, in which case the asteroid would explode in a blinding flash.

It has everything to do with distance as you can see in the formula g=G*M/R2 . If the moon became a black hole it would still have the same mass. Right now the moon has a radius of about 1000 miles which equates to an acceleration due to gravity of 1.6 m/s squared at the surface. If the moon were a black hole its radius would be tiny and you would be able to get a lot closer then 1000 miles to the center. If you got within 1 mile of the center your acceleration due to gravity would be much much higher. For example radius squared at the surface of the moon is 1000×1000 = 1,000,000. The equation above would be divided by 1 million to get the 1.6 m/s squared. If you were 1 mile away from the black hole the radius squared would be 1 mile times 1x1=1. Then the acceleration due to gravity would be divided by 1 instead of 1 million and your acceleration would be a million times greater that where the surface is now.

Right now, it already eats a fair amount of mass, as tiny meteors (and occasional big ones) rain down on it every day. Once it became a black hole, it would eat far less. Gravitationally, meteors would behave exactly the same while they're outside the moon's former radius. This is because the gravitational field of a spherical shell is exactly the same as for a tiny object of the same mass. And a body like the moon can be seen as a series of concentric spherical shells. This makes calculating orbits very simple, because you can treat a planet as a point object of the same mass.

Once the moon becomes a black hole, any meteors that would have crashed into its surface are now more likely to just whiz by the black hole, then continue on their orbit of the sun. Only a few, which are headed straight for for the smaller black hole at the center would be at risk of getting sucked in. Meteors that just pass near it, even within a few miles, would undergo a different deflection than predicted by Newton's theory, but probably wouldn't get eaten. Meteors would pretty much have to hit it directly. It would swallow up a tiny portion of the solar wind, but that wouldn't amount to much at all. Your basically asking how much matter currently gets eaten by a given 0.01 square millimeters of today's lunar surface.

About 32 femtograms per day

As others have explained, a black hole doesn't really suck in matter. It uses gravity to pull matter in, but that is not any different from how the moon acts now. Only matter that is on a direct collision course with the black hole will actually get eaten up, any other matter will just pass by and fly off into space again.

A black hole with the same mass of the moon would have a frontal area about $3.4 * 10^{21}$ smaller than Earth. So our black hole would also get hit by $3.4 * 10^{21}$ times less particles. Earth gets hit by about 400,000 tons of interstellar dust per year. Combining these numbers would mean that a black hole the size of the moon would eat about $32$ femtograms or $7.1 * 10 ^{-17} lbs$ worth of interstellar dust per day.

Probably much less than its present rate.

Moon has a lot of surface to "catch" meteorites and convert their kinetic energy into heat (so they won't bounce back into space). If made a black hole, most of these will pass in a parabolic/hyperbolic orbit around the black hole and leave for good.

In order to get something into the black hole (less than a 1mm for the Moon case), one has to target it pretty much exactly. If the object is a larger solid body, it will be converted to sand/dust by tidal force and most of it expelled back in pretty much all directions - all particles going in their very own parabolic/hyperbolic orbits.

p.s. I am not sure how much of actually sucked matter will convert into radiation. Larger black holes are pretty good and cocnvert 20-40% of the mass into light, that's how quazars work. If so, a collision with an ordinary meteorite could be pretty spectacular.