Most probably, we'll be able to mass manufacture diamond nano-threads and carbon nano-tubes in the near future, but they might not be strong enough. Is there a way to significantly reduce the tensile strength requirements? We also need to take heavy payloads, on board safety and maintenance equipment and climbers into account.
The simplest way to reduce the tensile strength requirements of a space elevator is to taper the cable instead of using a constant cross section.
First off to start thinking correctly about a traditional space elevator, you have to remember it is literally hanging down from orbit and all the weight of the cable is supported from above. The problem with a space elevator is that at some point a cable of any material is going to weigh too much to support itself, the concept is known as breaking length. In choosing materials for this application you try to optimize these factors: low density (light weight) and high tensile strength, to get the longest breaking length possible.
If the breaking length is not long enough to reach to geosynchronous orbit (or beyond for the counterweight) for a traditional style space elevator you can lengthen the maximum length possible by using a tapering cable.
Imagine the following scenario, I have an allow steel cable ~8 km long (this is the breaking length) supported from the top. If I add any more cable to the bottom, the increased weight will break the cable. How can we make it longer? Take the original cable and cut it in half, you now have 2x 4 km cables held together. Each shorter cable can support the weight of a 4km long section of cable at the bottom, or 8 km cable total together. So by tapering the cable I can now build a steel cable that is 12 km long that won't break under its own weight. This tapering can continue to lengthen the maximum cable at the expense of requiring more and more material to support it.
This is of course a simplified example, assuming a constant gravitational field over the length of the cable and neglecting the centripetal forces acting on a space elevator (both of which should actually increase the breaking length of a material).
For a very detailed discussion of the use of space tethers, including the effects of tapered cables in non-uniform gravity fields, read the book.
I asked the question above previously and found that if you have your space elevator not at GSO but at a lower orbit and traveling and it was slow enough to be caught by an airplane you could reduce the total length of the elevator by 30%. This sounds like a lot but is still 25,744 km but better than 35,800 km (plus extra to counter balance the elevator.) This reduced length would reduce the tensile strength needed for the elevator.