So an interesting thing about headphone cables is that the general principle is not too different from speaker cables, which have of course gathered quite a bit of "controversy". What they both boil down to, though, is Ohm's Law. To try and put it simply, there are essentially three factors involved:
1. A given impedance of your speaker/headphone (in ohms)
2. Resistance in the cable which increases the thinner and/or longer it is (also in ohms)
3. Potential output of your amplifier (in volts)
Due to the law, your total power that goes out to the speaker/headphone depends on all three of them. The less total resistance/impedance, the more power will get through. This is why 2 and 4 ohm speakers are easier to drive than 8 ohm speakers, and same goes for headphones where 32 ohm impedance is easier than 300 or 600 ohm impedance. The purpose of a cable is to provide as little in-between resistance as possible, as significant additional resistance will reduce the power that the amp is able to supply. Thick gauge and small lengths are needed for small impedance with large power outputs. There are plenty of numbers already floating around for speakers, such as
this excellent article by one of McIntosh engineers.
For headphones, the same effect applies. However, the required power to drive headphones is far less than what would be needed for speakers. Moreso as per the above, the required cable thickness goes /DOWN/ as the impedance goes /UP/. Think of the headphone impedance as an obstruction at the end of a hose - you can increase the size of the hose all you want, but since the flow through it remains the same (what the amp supplies), your total output flow does not change.
The resistance of a wire of a given gauge and length is easy enough to calculate for standard copper, and there are plenty of calculators for that. It's recommended that the total resistance of the wire be no more than 5% of the system impedance. The link above just about halves that even further on the safe side. For HD800, which is 300 ohm impedance, you'd want a cable that has a total resistance of no more than 7.5 ohms. As per
this calculator one can easily get away with using 30 awg wire (VERY thin) over a distance of 50 feet (lawl). So no, thicker cables (assuming they have thicker copper) won't do anything for headphones outside of some truly bizarre cases like running a cable over a few miles.
TL;DR: Due to basic physics, premium cables that are marketed as means of improving sound for either speakers or headphones are complete bunk. There is however some value in having nice cables that are more durable or flexible, but those can be had at a modest price increase over lamp cord.