If the Moon were suddenly transformed into a tiny black hole with the same mass, it would continue to orbit the Earth at the same distance. Ocean tides due to its gravity would continue normally. There would not be much effect except that it would no longer be visible with the naked eye and would no longer reflect the sun's light back to Earth. If you found it in a telescope, you might see gravitational lensing as it passed in front of the star field. Objects like probes or old spacecraft stages orbiting the Moon would continue to do so.
The only danger would be that if things fell into it I suppose you might get dangerous X-ray and gamma ray emissions from its accretion disc that would be a problem at such a close range. That would not be an issue with a primordial black hole much further away.
If there were such an object we could send probes to orbit it and study it, and some experiments may involve firing objects or shooting lasers or beams of particles into it to attempt to learn about the quantum effects at the event horizon. This could be massive for physics, allowing us to access and observe conditions and energies not replicable here on Earth with any current technology.
BTW we don't have any hard evidence that primordial black holes exist, but many theories predict them. So far such predictions around black holes have a pretty good track record. If you made me bet, I would bet on them existing. They are a candidate for some or perhaps even all of dark matter, though even if that's not the case they might still exist. It's possible that the dark matter haloes we can spot with gravitational lensing are clouds of these things. ("Clouds" of course is a misnomer-- the distance between them would be many light years.)
If planet nine is a PBH it means that at some point one was captured by our solar system into a Kuiper Belt orbit. Even if planet nine isn't one, there still may be small asteroid mass PBHs in our solar system, so we still might find one. They would require extremely sensitive X-ray or gamma ray telescopes or highly accurate gravitational models of the solar system to detect.
