Yes and no - Edit 1
Before modification by Isaac at 22/11/2011 09:21:04 PM
The material has no obvious use in the connection cable, but wouldn't an ultra light structural material for the orbital platform at the top of the cable greatly reduce the required tensile strength of said cable?
Meaning: What if that aircraft carrier only weighed as much as a speed boat?
Unfortunately the station at the top of a space elevator happens to represent that exception to the rule about lighter being better for space launches, most designs call for a big dumb object at the end to serve as a counterweight, so there's not any particularly special advantage to a low-density building material for the nominal terminus station, its mass is essentially determined by the cable.