I've never actually attempted this, but I believe the solution you're looking for is called a spline IK. What I think this does is bend a spline along a bone rig "chain". I could be wrong on this though, but it might be a good starting point.

What I'd do is just make a really thin cylinder with lots of vertical segments. Then just make a long chain of bones going from the top of the cylinder to the bottom. Select both the bones and the cylinder and do a Skin/Soft Bind. Create a handle between the first and last joints. Now when you raise and lower the handle the chain should go slack to some degree.

If you're having trouble getting the chain to slack in the right direction, you can always pre-curve the chain a small amount during rigging, and then just overstretch the handle to make the rope go straight during animation.