Let's say you want to have the wire run only into the left cup.
- Disassemble the headphones
- Remove the wire from the right driver.
- Figure out a way to run a two conductor wire from the left cup, over/inside the headband, to the right cup.
- Feed the old right wire into the left cup and splice it to the new bridge wire running from left to right.
- Solder the bridge wire to the right diver.
Electrically, that will work. The purists would want you to make the resistance and length of the left and right cables the same, which might require you to add to both sides, and then figure out how to hide the extra cable on the left side. Aesthetically - it might be OK, or it might look like cr@p.
Consider a headphone with a very simple headband, like the Grado SR60. How would you run the cable from the left to the right over the headband? The headband is just a simple, bent, flat piece - it's not hollow. You would need to use some sort of headband cover or something.
What headphones are you talking about?