I am writing a script for science fiction RPG where everything takes place in a science station orbiting a black hole just outside the Schwarzschild radius. But it occurred to me I have no idea what kind of scientific experiments or observations would need such a station. We can't receive any data from the blackhole anyway.

So, would such station have a purpose? What kind of experiments could scientists perform there that they couldn't anywhere else in Space? What kind of (current) hypothesis could they test? What kind of observations they could perform? Does the kind/size of black hole matter in this respect? Thanks a bunch!

Hawkins radiation observation

Stuff does come out of black holes.

Hawking radiation is blackbody radiation that is predicted to be released by black holes, due to quantum effects near the event horizon. It is named after the physicist Stephen Hawking, who provided a theoretical argument for its existence in 1974.

Lots of science and new discoveries to be made from those.

High energy experiments

Ditch those particle colliders on Earth. Some black holes have accretion discs that output a lot of energy.

(...) TON 618 is believed to be an accretion disc of intensely hot gas swirling around a giant black hole in the center of a galaxy. The light originating from the quasar is estimated to be 10.4 billion years old. The surrounding galaxy is not visible from Earth, because the quasar itself outshines it. With an absolute magnitude of −30.7, it shines with a luminosity of 4×1040 watts, or as brilliantly as 140 trillion Suns(...).

But mainly, material science

(...)a science station orbiting a black hole just outside the Schwarzschild radius(...)

Your station is probably made of unobtanium.

In astrophysics, spaghettification (sometimes referred to as the noodle effect) is the vertical stretching and horizontal compression of objects into long thin shapes (rather like spaghetti) in a very strong non-homogeneous gravitational field; it is caused by extreme tidal forces. In the most extreme cases, near black holes, the stretching is so powerful that no object can withstand it, no matter how strong its components(...)

So you may as well play challenge accepted with physics. You will also need the station to generate a handwavium field to keep people and stuff in it from breaking into subatomical particles.

All Things Gravitic

I'll have to find the researcher who suggested that fluctuating space-time can drastically reduce the energy requirement for gravitic manipulations, such as the Alcubierre FTL drive.

Near a steep, rapidly fluctuating gravity well is exactly where you want to test that. If true, it could result in FTL (or next-generation more power efficient FTL), artificial gravity, gravitic communication, gravitic computation, gravity lasers. In short, all things gravitic.

Quantum Mechanics

And it's important to remember that, from a materials perspective, a singularity is a rapidly spinning grinding wheel composed of soft gel at the far edges and impossibly rock-hard, razor-sharp, space-time as you get closer to the singularity itself.

It might not be necessary to build a huge accelerator to bust subatomic particles. Instead of accelerating particles to near the speed of light and driving them into one another, shoot a particle towards the event horizon at an angle and let the event horizon rip it to shreds. Use an angle sufficiently shallow that the results of the interaction are observable.

Spontaneous Generation of Matter and Anti-Matter

You could observe the spontaneous creation of matter and anti-matter to learn how to produce either or both on-demand at industrial scales.

Looking Inside Color Confinement

At these high-energies it may be possible to look at quarks inside the color confinement properties that keep them quarks from being isolated and studied individually. We suspect a great deal about how the strong nuclear force works, but a lab on the edge of a singularity would open up the ability to test these hypothesis cheaply (it could be done with a large enough terrestrial collider also)

Mass Manipulation

You could use this to perform studies of the Higgs field, and what causes things to have mass. How mass seems to be coupled to gravity can be confirmed and studied in greater detail.

Negative Energy and Negative Mass

If negative mass or negative energy particles are spontaneously created and destroyed in some small amount, a lab on the edge of a black hole is the place to look for them.

The Granularity of Space-Time

So far, all experiments done have indicated that while electro-magnetic and strong/weak nuclear forces have quantum behavior: that is everything has discrete behavior - one unit, two, three...; but space-time itself has appeared in every experiment to be smooth. It may be possible near a singularity to test this smoothness more rigorously. I'm not sure what quantum space-time would prove, but it would change a lot of things.

Compactified Dimensions

It should be possible, under the extreme energy conditions, to construct at least a few experiments to probe compactified dimensions. The math predicts either none, 10, or 11. How many are there, really? Do they exist at all? What does it mean (new ways of storing data at higher densities, communication, travel)?