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.
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)
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.
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)?
Faster than Light Travel / Communications
If the world you are building does not yet have it, all theories of faster-than-light travel assume some compactified dimension of space-time that can be exploited ("hyper-space" literally "extra space") to skip over the places in between Point A and Point B. If you can prove compactified dimensions, you can determine if such an exploit is possible. Or, since the dimensions are compactified (small), they might not provide any such skip-over (but might have other interesting properties).
The black hole may not have consumed all of the matter near it. The behavior of high-energy plasmas in the accumulation belt may be interesting. There may be all sorts of new elements and chemicals in the far-flung debris field away from the hole. Some smaller (asteroid-sized) chunks of degenerate neutron-star like material may have been ejected in the birth of the black hole, and available for capture and study.