The yellow/blue three strand stuff is polypropylene. It's cheap, which is a plus, and easy to find. I'd probably be tempted to start with that (if I didn't have access to thousands of pounds of rope
) and then switch it out once the thing was built if it looked like doing so would give me any real benefit.
A CanDo said, raw nylon is soft and white, and quite stretchy. It would probably be your best bet, and you could dye it (cheap fabric dye - eg: RIT). Might be a little harder to find, and it will be more expensive. I'd stay away from the polyesters - dacron etc. and anything remotely "high tech". If your ropes don't have abit of elasticity you are basically making a giant wooden-frame-breaking-machine.
None of the synthetics will have too much problem with set in this type of application. They will all stretch in a bit with use, but will quickly get to a point where you get back virtually all of the energy that you put in.
Matthias