Woodchops. Not that obscure in themselves, but I've never seen anyone doing them the way I do them. The way I do them, they're right up there in my favourite core exercises, but I haven't had regular access to a cable tower to do them in about 18 months.
Sit ups. Again, not obscure in themselves, but I don't see people using the technique I use on them. The way I'm used to seeing people do them (if they ever do, which has been rare for the last decade or so due to the widespread belief that using your hip flexors means not using your abs) allows you to do a lot of them without heavily using your core. The way I do them makes it hard for me to do 10 and gives an intense ab contraction, while amplifying how much hip flexor is used (so, instead of trying to make them more ab-dominant by removing the hip flexor work, I increase both elements in a way that I've found quite effective).
Lower trap static flys. I've only known a handful of people who've done these, and each of us has gotten strange looks in the process. I do them with epic 1.25kg weights, and it's hard. But they do a really good job at establishing a mind-muscle connection with the lower traps, which in turn makes all back work better, endorses good posture and helps with thoracco-scapular tightness for squats and bench press.