It's well understood that the moon has an effect on how much the earth spins. My understanding is that if the moon was bigger, it could impact tidal forces to a larger extent, and possibly slow the rotation of the earth.
An article by space.com titled "This Huge Black Hole Is Spinning at Half the Speed of Light" describes that black holes could be rotating at 50% the speed of light.
Can we tell if the speed of a black hole's rotation is impacted by the amount of matter orbiting it?
Best Answer
Short answer: Yes!
Spin (angular momentum) of a black hole could be changed by interaction with the surrounding matter in a variety of ways.
The laws of black hole thermodynamics would be respected in such processes. This, in particular, means that the area of the event horizon will always be increasing even when rotational energy is being extracted from the black hole. Also, angular momentum of black hole would satisfy $J<GM^2/c^2$ (where $M$ is the black hole mass), it is impossible to spin up the black hole beyond this bound.
If a body plunges into a black hole its angular momentum (mostly angular momentum of its orbital motion but also any intrinsic angular momentum it had) becomes part of the black hole's. So if a black hole is surrounded by accretion disk, then transport of angular momentum within and into the black hole is an important aspect of processes happening there and this transport would occur via a lot of means: plasma turbulence, magnetic fields, electromagnetic and gravitational radiation etc. Over the course of its evolution accretion could “spin up” black hole quite close to maximum possible values of angular momentum (for a given mass).
It is possible to transfer angular momentum (and energy) to a black hole via tidal effects see e.g. this paper. For example, this would happen if a black hole is part of a binary (with another black hole, neutron star etc.). This is the direct generalization of Earth's rotation slowing by the tidal influence of the Moon as mentioned in the OP. When placed in identical environments, a rotating black hole absorbs more energy and angular momentum from tidal effects than a nonrotating black hole. But even for rotating black holes under most circumstances this type of mass/angular momentum absorption is too small and could be usually ignored.
Rotating black holes exhibit superradiance, a phenomenon when the flux of radiation (electromagnetic or gravitational) impinging on a black hole is amplified. This effect is conceptually similar to Penrose process only for waves rather than particles. The necessary energy and angular momentum carried away is supplied by black hole's rotation, and it slows down as a result. A related concept is the black hole bomb, a runaway superradiant process.
Finally, special mention should go to Blandford–Znajek process which extracts rotational energy (and thus slows down the black hole's rotation) via magnetic fields from external sources.