Is it possible to have a lightsaber that bends like a whip would? If so, how?

No, closest would be to model a new saber hilt, with a glowing cloth attachment to simulate the blade.

Actually, yeah--I've always wondered if it could be done by attaching something like Aayla's tentacle to the end of a saber hilt. I don't really know how one would go about doing that and all...

It probably could be done if one were to make all new animations.

Aayla's lekku aren't "tentacles," per se. They're part of her model and not addon meshes. You could attach a Wookiee tentacle onto a saber hilt but ideally you'd just model a new tentacle string.

Could i attach the wookie tentacle on and then just make the blade transparent?

Yes, you could. But the invisible blade wouldn't always line up with the tentacle.

It would be close enough. How would I attach it? I understand you can have addon geometry attached to a certain joint, but a lightsaber has no joints.

"Joints" == hardpoints. Lightsabers have an hp_fire.

I always figured you could just make a saber throw that's short range and fast.