Can work hardening of a metal be avoided? My left earbud recently broke mid-wire: the bit that I like to fiddle with and bend. I fixed it, but I was wondering whether there are metals that don't work harden, or resistant to it? 
Is there a way that you can treat normal metal to prevent it from, or at least reduce the effects of, work hardening. 
Can these be used in headphones commercially? (i.e are they economically viable?)
 A: Work hardening does not cause things to break, but in fact will cause them to resist further plastic deformation increasing their strength. Wires bent back and forth may eventually break due to fatigue. The material at the edge is compressed and stretched resulting in fatigue. How much cyclic stress there is determines how many cycles the material can last its fatigue limit.
From that article:

If the stress on the wire at the edge is below the endurance limit then the wire could be  bent back and forth indefinitely. One way to reduce stress and allow for more flexible wires is to use braided wires. This works by reducing the cross section of each strand and thus reducing the amount of strain required to produce a certain bend radius.
Of course, if by fiddle with and bend you mean deform past the elastic limit so that there is a kink, then you are inherently exceeding the yield stress every cycle and micro cracks will propagate very quickly. If this is what you'd like to due to your cords and would like a cord to survive this treatment, I would design such a cord with thin braided wires that were surrounded by a self-healing polymer. This would allow you to plastically deform the cord and have the cord heal itself.
As for metals that can recover from being yielded there are shape-memory alloys, though I think they would be cost prohibitive.
A: 
My left earbud recently broke mid-wire: the bit that I like to fiddle with and bend. I fixed it, but I was wondering whether there are metals that don't work harden, or resistant to it?

The simple answer is No. Work hardening is result of plasticity. There are no perfectly elastic materials 

Is there a way that you can treat normal metal to prevent it from, or at least reduce the effects of, work hardening.
  Can these be used in headphones commercially?

As someone pointed out , Annealing is majorly used to soften a material so that its more workable. So essentially if you want to reverse the effects of work hardening this is the only route that is viable.
Work hardening leads to essentially hardening or strengthening till no further ductility is produced or material fractures or fails. If the material doesnt work harden or if there is no  plasticity it is going to fail by brittle fracture at yield point.
The wires you mention can be produced/manufactured only if they show plasticity(or work hardening) in the first place, if they were perfectly elastic they couldn't be used due to risk of brittle fracture or low toughness. 
A: work-hardening doesn't persist if the service temperature is above 1/3rd to 1/2 of the melt temperature- at which thermally-activated diffusion effects are sufficiently strong to relieve the dislocation pileups via dislocation climb. a thick lead wire cannot be work-hardened at room temperature because its melt temperature is in this range- but neither can it be used as a spring to help hold the earphone in place, for this same reason: it creeps significantly at room temperature.
A: Just to add to the other answers, you unfortunately will have to replace the whole wire/earbud set. Melting the wire will only fuse the different isolated wires that exist in one cord, and you will have an at best dysfunctional if not dangerous product that works in only one ear. Sorry to be the bearer of bad news.
