As the title says, what happens when a gravitational wave approaches a black hole? I would presume that something interesting happens because of the way spacetime works near black holes but I have no knowledge to back it up.
No, gravitational waves cannot pass through a black hole.
A gravitational wave follows a path through spacetime called a null geodesic. This is the same path that would be followed by a light ray travelling in the same direction, and gravitational waves are affected by black holes in the same way that light rays are. So for example gravitational waves can be refracted by gravitational lenses just as light waves are. And just like light waves, if a gravitational wave crosses the event horizon surrounding a black hole it is then doomed to travel inwards to the singularity and can never escape.
There is one caveat to this. When we talk about a gravitational wave we generally mean a ripple in spacetime that is relatively small. Specifically it is small enough that the energy of the gravitational wave does not significantly affect the spacetime curvature. So when we calculate the trajectory of a gravitational wave near a black hole we take the black hole geometry as fixed, i.e. unaffected by the wave, and we compute the trajectory of the wave in this fixed background.
This is exactly the same approach as we use for calculating the trajectories of light rays. Since light rays carry energy and momentum then, at least in principle, they have their own gravitational fields. But for both the light rays and gravitational waves likely to exist in the universe the energy carried is too small to make a significant contribution to the spacetime curvature.
When you say in your question:
I would presume that something interesting happens because of the way spacetime works near black holes
I would guess you are thinking that the gravitational wave could change the geometry near a black hole, but as described above typical gravitational waves don't have enough energy to do this. It would be reasonable to ask what happens if we give the wave enough energy, but the answer turns out to be that it no longer behaves like a simple wave.
Gravitational waves exist in a regime called linearised gravity where they obey a wave equation that is basically similar to the wave equation light obeys. If we increase the energy so much that gravity becomes non-linear (as if the case for black holes) then the oscillations in the spacetime curvature no longer obey a wave equation and need to be described by the full Einstein equations. For example it has been suggested, but not proven, that really high energy gravitational (or light) waves could interact with each other to form a bound state called a geon. I confess that I'm unsure how much work has been done studying oscillations in this regime.
Gravitational waves should be lensed by massive objects in a very similar way to light.
Light rays (and by extension, gravitational waves) from a distant object, that pass within 1.5 times the Schwarzschild radius (for a non-spinning black hole) have trajectories that take then towards the event horizon. Waves on such trajectories cannot escape from the black hole, so the basic answer is no, gravitational waves cannot "pass through a black hole".
However, far from "hiding" a source of gravitational waves, an intervening black hole would cause the presence of lensed and magnified images. For perfect alignment of source, black hole and observer, there would be an intense "Einstein ring" at an angular radius that depends on the relative distances of the source and the black hole.
Of course gravitational waves cannot be imaged at present, so what would be detected is an abnormally strengthened gravitational wave signal.
All the above is in the geometric optics limit that the wavelength is small compared with the lens. If the black hole is small enough (which depends on its mass), or the gravitational wave wavelength is large enough, then the behaviour ought to be analogous to a plane wave encountering a small, opaque disc (Takahashi & Nakamura 2003).
In which case we would get a diffraction pattern and perhaps a "bright" Arago spot in the centre, though I am not aware of any such calculations in the literature.
This is not an unlikely scenario. For example, the gravitational waves detected by LIGO have relatively high frequencies of 10-1000 Hz and therefore wavelengths of 30,000-300 km, which are as big as the Schwarzschild radii of 10,000 - 100 solar mass black holes and certainly bigger than black hole remnants of stellar evolution.