Oh hey, I'm an EE, and I do wire harnesses for a living (in heavy trucks)!
...okay, it's not really audio...Still, the general practices can apply.
First off, make sure you have the right gauge. Larger wire (lower gauge number) has lower resistance. The line resistance will sap power as losses in a DC environment, but it can also mess with your reactive devices by altering the impedance relationship between amp and speaker. The Speaker Wire page was already linked if you want more info. With headphones, it is very easy to stay within acceptable ranges.
Second, harness (or cable) construction. This is really about convenience when it comes to headphones (at least IMO). With wire harnesses, you want to minimize strain on the wires, and that sometimes means allowing for a certain degree of movement, especially between suspensions. You also want to consider what kind of wear can happen to the wires and shield appropriately. Take a took at a heavy truck some time. You'll see the chassis harness surrounded by convoluted tube, and if it passes near exhaust, you'll see a heat wrap. You can apply this lesson to headphones by choosing a cable construction that meets your needs and environment. Certain outer materials are going to keep the cable "slick" relative to most surfaces (if you're moving it around), and certain materials will generally be more rugged, resistant to damage.
Finally, routing. This has a bit to do with cable/harness construction, but it's probably a better lesson for interconnects. Just be mindful of where the cable needs to go and how it's going to get there. Shorter is generally better so you don't accidentally get snagged. You might still need some length if your components are going to be far apart, and then you should consider ways to keep your cables from getting lost in the rat's nest of wires behind your components. Actually, you generally should try to prevent a rat's nest from forming.
So the primary reason, IMO, to use "better" cables in audio really come down to non-audio needs. The stock cable will probably sound just as good as a "better" one (assuming that designing engineer is earning his/her salary), but you can definitely put a price on convenience (not $1,000+, of course).