The gravity near the black hole and event horizon would break any string. Even without any mass at the end, the string would tear itself apart at some point.

The mass of the string itself would cause it to accelerate towards the hole. And as you get close the acceleration would approach infinite and the speed of the end of the string would approach the speed of light. And there is no string that could possibly take that kind of strain. In fact, as you get really close the gravitational force will overpower the strong force and the individual atoms will come apart.

If you did manage to get an event horizon separated from a black hole it still wouldn't work. Anything moving towards the event horizon still requires an infinite time to cross it. The event horizon is a two way barrier, nothing on the inside can effect us and nothing on the outside can effect the inside. Thus the string question is meaningless.

Jay