Fine stranded multi core wire (i.e lots of very fine wires in a bunch, instead of fewer heavier wire) are the most flexible . Many R/C shops have this with silicone rubber outer isolation and it comes in many gauges - used extensively for probe wires in pro multimeters as well.
Another method, for sharing the load over a length of wire, besides the U-shape, is to make a single coil loop (just be careful that you don't get interference as a result).
The best solution, if at all possible, is to use slip rings (or a similar setup) where the wire would bend. They're not cheap, but they last and if you are just a bit handy, you can make them yourself.