I don't think this would work because you're black holes would be too far apart and provide too small a change in delta-V to be interesting in a sport.
Black holes are generally formed by collapsed stars, and as such they tend to be as far apart as solar systems, ie very far apart. Even if you have technology making black holes closer then would naturally form they would still have to be very far apart or they would suck in each other and collapse into one larger black hole. this means you will have, at most, one black hole close enough to you to have any appreciable gravitational affect on you.
Before anyone says it yes in theory you could on rare occasion have two black holes in stable orbit of each other, but even then they have a minimum distance far enough apart that likely only one will be applying significant gravitational pull at a time.
However, this black hole will not have an interesting affect at a rate worth calling someone a daredevil, though why depends on your FTL travel.
- If you are staying at sublight speeds then you're craft is moving so slow that it would take days to approach the black hole, any maneuver that takes days to complete probably isn't going to make a very fascinating sport.
- Most viable FTL options involve not moving in 'normal' space and thus would not be likely to be affected by gravity.
- If you have FTL travel fast enough to make regular trips between star systems a regular occurrence that doesn't take a decade to complete then you are moving so fast that the you zip by the black hole too quickly to notice it's gravity. The only daredevil sports you could pull off is "how close will that idiot get to the black hole without running into it", but this is really just a game of chicken, no different then sticking a giant pole in the middle of a racetrack and seeing how close your racecar drivers can pass it in a lap without hitting it.
- You could, in theory, create a third type of FTL travel, too slow to make regular trips between systems viable but fast enough to make it possible to approach a black hole fast enough to make it a viable sport. However the delta-V provided by gravity would still be too small relative to the speed your craft was going to have a significant effect, your engines would be able to push you forward at a rate dozens of times faster then the blackhole altered your speed. Furthermore to get close enough to the black hole that it can impart any remotely interesting amount of delta-v for the short length of time you would be near it you would be so close the gravitational shear affects would likely tear you're craft apart.
- A gravity well, like a black hole, will not add speed in the long run, it could in theory help you change direction, but total speed will not be increased by approaching near one and then departing it.
Basically I see no means that black hole dare-deviling makes sense in realistic physics. Luckily as a writer you get quite a bit of suspension of disbelief which allows you to make up your own rules. I'd suggest you create an intra-system handwave FTL that happens to behave in a manner similar to what you want to happen because insert quantum tachyon technobable. First you would need to have two FTL systems, one for traveling between solar systems that was faster and one for intra-sytem travel which is fast enough to allow rapid approach to planets but not too rapid of an approach. Since you are making up a handwave FTL system anyways simply say that this one is explicitly affected by local gravity in a manner more drastic then would happen in if you were traveling in normal space at sublight speeds. In other words your not Just being accelerated by the gravity of a black hole, which doesn't provide enough delta-V to be interesting at those speeds. Instead just getting closer to a massive gravity well allows your intra-system FTL to accelerate your craft faster because quantum magic. Similarly you can have practically any affect you want happen if they get too close to the black hole, since your pretty much making up the physics of your FTL drive anyways.
Of course even then there is another problem, black holes really aren't that different then any other gravitational body. Other then being super-massive the way the gravity work's doesn't change. If you replaced our sun with a black hole of the same mass all planets would orbit in just the same way. Given that fact any massive body could provide the same gravitational pull. It makes more sense to have your dare devils zooming past planets then black holes, they will be closer together and if you just approach them a little closer then the black hole you would still get the same overall gravitational pull on you. That is unless you do more handwave magic about how the density of the gravitational object affects how your FTL system works.
