What do black holes spin relative to? What do black holes spin relative to?In other words, what is black hole spin measured in relation to?

Spinning black holes are different from non-spinning black holes.  For instance, they have smaller event horizons.  But what are the spins of black holes measured relative to?
Let's first look at an example with common objects.
Example
Let's say there's a disc on a table that rotates at 60 rpm.  When you are standing still it spins at 60 rpm. But if you start running around it, it will move faster or slower relative to you.  In this case, the disc has a ground speed, 60 rpm, because it has something to spin in relation to, in this case, the table.
Black Holes
Now, let's say that there is a spinning black hole. Because there is no control for the black hole to spin relative to, its spin must be relative to an object, for example, you. If you stand still, it spins at a constant rate.  But if you start moving around the black hole in the same direction as the rotation, according to Newtonian physics, the black hole would spin at a slower rate relative to you.  Since a faster spinning black hole has a smaller event horizon, in the first case, there would be a smaller event horizon.
Then how do scientists say that there are spinning and non-spinning black holes? Is that just in relation to Earth?
Ideas
First Idea
My first idea is also one that is more intuitive. When I move around the black hole, the black hole spins slower relative to me and consequently has a larger event horizon.   In this idea, the black hole would just behave like a normal object.  This would mean that if you went really fast around a black hole, you could get a lot closer to the black hole that if you were standing still.
This is kind of like a satellite that orbits Earth. The slower it moves, the easier it is to fall to the Earth. (I know this is a horrible analogy.)
Nothing special happens here.
Second Idea
My second idea is that when you move faster around the black hole, the relative rotational speed of the black hole doesn't change.  Because of how fast it is/how dense it is and special relativity, moving around the black hole doesn't affect its speed.
This is like trying to accelerate past the speed of light.
No matter how much energy you spend, your speed barely changes.
I don't understand how this one would work.  Why won't the rotational speed of the black hole stay the same?
Conclusion
What do black holes spin relative to?  And what happens if you move around it?  There are lots of questions that ask how black holes spin, or how fast they spin, but as far as I know, none of them address this question.
 A: This is just Newton's bucket in modern garb. The best explanation of this effect that I have seen is in Carlo Rovelli's book Quantum Gravity, which explains it as rotation with respect to the gravitational field. According to Einstein's Theory of General Relativity, the gravitational field is a real physical entity. And Rovelli says about Newton's bucket (on page 56 of the 2005 hardback edition):

Einstein's answer is simple and fulgurating:
The water rotates with respect to a local physical entity: the gravitational field.

Rovelli regards this as so important that he underlines it, as well as putting it in italics; but my formatting skills don't run to that. And yes, fulgurating is a real word.
A: 
What do black holes spin relative to?

Relative to an inertial reference frame infinitely far from the hole, in which the hole has no translational motion.

And what happens if you move around it?

A spinning black hole is azimuthally symmetric. It “looks” the same from any azimuthal angle. Its spin parameter $a$ in the Kerr metric has nothing to do with how fast you move around it.
A: 
But if you start running around it, it will move faster or slower relative to you.
In this case, the disc has a ground speed, 60 rpm, because it has something to spin in relation to, in this case, the table.

Actually, this is fundamentally incorrect. The spinning of the disk has nothing to do with the table in principle. Acceleration, including spinning, is not relative. It can be measured without reference to any external object. For example, using a ring interferometer, or a gyroscope.
It does not matter if the object is a disk or a black hole or anything else, spinning is not relative like inertial motion is.

When I move around the black hole, the black hole spins slower relative to me, and consequently has a larger event horizon.

The event horizon is a global and invariant feature of the spacetime. Your motion does not change it. Of course, you can use whatever coordinates you like and make the coordinate size change as you wish. However, which events are on the event horizon is unchanged by your motion.
A: 
Let's say there's a disc on a table that rotates at 60 rpm. When you are standing still it spins at 60 rpm. But if you start running around it, it will move faster or slower relative to you. In this case, the disc has a ground speed, 60 rpm, because it has something to spin in relation to, in this case, the table.

No, the table is not necessary to observe a difference between a spinning disc and a stationary disc. If you are rotating with a spinning disc, and are using a reference frame in which the disc is stationary, that reference frame is not inertial. There will appear to be a "centrifugal force" pushing you away from the spinning disc, and to keep next to it, you will have to have a force pushing you towards the disc. So you can tell the difference between a rotating disc and a stationary disc because you can be in a co-moving reference frame with a stationary disc without a centrifugal force appearing.
Now, there is a phenomenon called frame dragging in which a rotating black hole will distort space-time around it. Near the black hole, this will reduce the apparent rotation. But far away from the black hole, frame dragging become negligible, and the black hole's rotation can be measured with respect to inertial reference frames.
The other answers saying that there is no need for there to be anything else to measure it with respect to are somewhat wrong, as frame dragging is dampened by the mass of the rest of the universe. If everything in the universe other than the black hole were to disappear, it would be impossible to observe the black hole as rotating.
A: You may think of this as an application of Mach's Principle. This represents an observed fact of physics and cosmology, not derivable from some other principle. The local, non-rotating frame appears to be determined by matter, mostly distant matter. General Relativity, partly inspired by this idea, covers the (small) effect of local matter, but does not demand Mach's Principle for the whole Universe. It's a proposition that has been tested to high precision.
A: 
Let's say there's a disc on a table that rotates at 60 rpm. When you are standing >still it spins at 60 rpm. But if you start running around it, it will move faster >or slower relative to you. In this case, the disc has a ground speed, 60 rpm, >because it has something to spin in relation to, in this case, the table.

The spinning black hole is a solution of the vacuum Einstein equation that describes the spacetime interval $ds^{2}$ which is an intrinsic property of the spacetime itself on which all the observers agree. In other words, what we call a spinning black hole is the spacetime itself, so the example described by the rotating disk on the table is not a good analogy.

Now, let's say that there is a spinning black hole. Because there is no control >for the black hole to spin relative to, its spin must be relative to an object, >for example, you. If you stand still, it spins at a constant rate. But if you >start moving around the black hole in the same direction as the rotation, >according to Newtonian physics, the black hole would spin at a slower rate >relative to you. Since a faster spinning black hole has a smaller event horizon, >in the first case, there would be a smaller event horizon.
Then how do scientists say that there are spinning and non-spinning black holes? >Is that just in relation to Earth?

The meaning of the spin is that any observer outside the black hole and sufficiently close (i.e. in the ergosphere) can not stand still. This is called the frame-dragging. The "least rotating observer" is a locally non rotating observer whose angular velocity, defined by an inertial observer at infinity is $$\Omega = \frac{d\phi}{dt} = -\frac{g_{t\phi}}{g_{\phi\phi}}$$
When we take the limit for $r \rightarrow r_{+}$ we have $\Omega \rightarrow \Omega_{Horizon}$.

What do black holes spin relative to? And what happens if you move around it? >There are lots of questions that ask how black holes spin, or how fast they spin, >but as far as I know, none of them address this question.

So the angular velocity of the horizon is the angular velocity of a local non-rotating observer (at the horizon) measured by an inertial observer at infinity.
For a detailed discussion, you can find more material on "General Relativity" by R.M. Wald.
