There are three options that I know of for stretchy conductive strings, none very good.
A much better alternative would be something like acoustics, ultrasonics. It can even go other than line-of-sight, if that's a problem to be solved. You don't say what stability is required, either short or long term. Although the speed of sound in the atmosphere is dependent slightly on pressure, temperature and humidity, it is relatively simple to install an additional calibration recevier at a fixed reference distance from the transmitter, and make what is effectively a comparison between the reference and the measured distance.
So, a review of the stretchy strings ...
a) Any metal wire, obviously steel has larger elastic range than copper. The change in resistance will be tiny, and easily swamped by the temperature coefficient. A wire like constantin will have a much lower tempco, and may be more useable.
b) A rubber tube filled with mercury has been used in the past for exactly this purpose, however its toxicity rules out its use in this sort of application these days. There are other less toxic metal liquids available, though they are pricey and difficult to obtain.
An aqueous solution of some suitable salt, measured with AC, say copper sulphate between copper electrodes may be possible, but temperature coefficient would still be a problem.
c) You can get silver-filled rubber cord, for making conductive gaskets. This has quite a marked tension/resistance response, but it is weak, and prone to creep. A broad equivalent is the carbon-filled plastic anti-static bag material. The same problems of weakness and creep apply.