Wow! That's the first thing that I thought of. This graphene "super paper" could be used for the portion of the elevator outside the atmosphere, with carbon nanotube-based ribbon being used for the portion inside the earth's atmosphere (where it would encounter moisture). Either that, or use it the whole way with a layer of protection for the portion inside the atmosphere.
A manufacturing process could probably be developed which would allow this to be produced in continuous sheets.
Gee, the first thing I thought of was light weight body armor. I need to go eat some breakfast....