Agree, silicone -- or neoprene. Also look for extremely fine stranding.
Example: a #0 welding cable is more flexible than THHN #6, despite being over twice as thick, because it's made from a great number of fine strands, and a flexible rubber jacket. It is more expensive -- not that that'll be so big a problem for your purpose, as you probably won't be measuring your wires by the pound!
There are also unusual constructions, that look like metal tinsel woven with fiber, used for small signal, high flex applications such as internal speaker connections, telephones, headphones, microphones and so on.
Depending on your setup, you may also be better off using relatively springy wire, which is self-supporting and relatively free from fatigue. The downside is, the spring force must be factored into your mechanical design, and arranged to bear in such a way that it does not disturb the measurement. In electrical terms, this is analogous to using a radio-frequency coil instead of a transformer (a high inductance) to convert voltages; the component's characteristics can't be ignored, but once factored out (in the case of the RF coil, by resonating it at some frequency), it works just fine.
Tim