I looked up more details and found this.

The paper itself isn't being presented until September, but from what I see, the gimmick seems to be running a tether far enough out (like your #2 scenario) that the center of mass moves the actual asteroid out of the collision path. Of course, the tether would probably hit. As you point out, it would take a big mass and one hell of a tether.

The best way of dealing with the asteroid threat:

1) Establish a location and tracking program. I like to think we can construct dedicated observation systems either on Earth or in space that can locate even small objects and figure out their trajectory at enormous distances. A better, but more expensive way, is to develop probes to go to the most serious targets, land on them (or to be more accurate, stick themselves to them) and become a science lab, and most important, a transponder system that can tell where the asteroid is down to the inch. The tech to do this is here now. If the probe's only job was transponder, it would use almost no power.

2) Develop a system that can divert asteroids when they're still far enough away that you can almost do it by hitting them with a ping-pong ball. Maybe the probes would have auxiliary ion engines. Of course, the probes would now become a hell of a lot bigger. Better to have hundreds of microprobes tracking asteroids, then send the diverter probes as needed.

With advanced enough robotic technology, you could land something on an asteroid that would "mine" chunks off the object and fire them into space as propulsion mass using mass driver technologies. This would be either solar or nuclear powered. You can run this propulsion system until something breaks down or you run out of asteroid, and you could divert even a large object this way given enough time.