This is definitely the most stupid idea of the century. To get up to that station 66,000 miles up in geosynchronous orbit, you would not only have to lift the object up, but also accelerate it sideways to get up the the higher rotational speed of the station..
Here on earth, the rotational speed at the surface is about 1000 miles per hour (one full rotation in 24 hours). At 66,000 miles up, the rotation speed would be about 17 times higher, or about 17,000 miles per hour. The difference between that and the surface speed would have to be made up by a rocket firing sideways as the object was lifted to the station 66,000 miles up. The total rocket energy expended in such a lift would be nearly the same as in a conventional launch.
An asteroid counterweight would “orbit” at a matching the rotation of the Earth’s spin. Centrifugal force would keep the cable taut and the asteroid stationary, on a direct line projected from the center of the Earth along the cable.