I have a cheap set of Phillips SHE3000 earphones with a broken wire on the plug. Since I was repairing another set of headphones, with the same problem, I bought an extra plug, just to try repairing them too.
Here's my problem - inside the cable for each channel there's not only the ground for it, but also some weird white hairs. It does not look like those would survive soldering.
What are those and what should I do with them?
They are just essentially string to help support the cable. You should be fine soldering the cable.
One thing to remember about the cables in headphones is that it is usually enamel coated copper wire. You usually need to heat it up to around 390*C to burn off the insulation before you can make a decent solder joint. The way I do this is to put a large blob of solder onto the end of a soldering iron, then push the end of the wire that needs to be tinned through the solder and pull it back out again. This usually neatly removes the enamel and tins the wire on all sides making soldering easy.
Bonus info, when more than one of the wires is broken midway along the cable, the easiest thing to do is the cut each of the wires at staggered locations so that when you solder them you don't need to worry about insulating one wire from another as the solder joints will be at different locations and the enamel on the other wires will be intact on the other wires. If you are soldering on a new plug though, you don't need to worry about that.
Headphone wires often get lots of mechanical stress,
for example if you drop them, while the other end of the cable is fixed somewhere.
I think these hair-like things are fibres of some very strong material, probably a para-aramid like Kevelar, the fibres you make bullet proof and chinsaw proof clothing of - see below.
They are working as strain relief when otherwise the copper wire would get stretched and break pretty fast. The tensile strenght of kevlar is about 30 times the one of copper stranded wire.
DuPont offers Kevlar directly targeted at headphone cables:
From Kevlar® Ropes and Cables | DuPont USA:
For more than two decades, DuPont™ Kevlar® brand aramid fiber has provided a lightweight, flexible, and dimensionally stable means of helping to strengthen ropes and cables used across several industries, from mountaineering ropes and fishing lines to electro-mechanical cables and fine gauge cables for electronic device applications such as mobile phone cables, computer power cords, USB cords, and MP3 earphone cables. Ropes and cables made of Kevlar® help deliver performance and value to customers in the fine gauge cable industry by providing excellent robustness, fatigue resistance, shrinkage, and durability. The fiber’s resistance to chemicals and temperature extremes makes it an ideal component for ropes and cables under severe loads in harsh environments, from the bottom of the ocean to the surface of Mars. Ropes made with Kevlar® brand fiber appear in a variety of constructions, including 4-strand, wire lay, braided and parallel yarns, and strand ropes. They range in diameter from 1mm to 250mm, and have been strength-rated up to 1500 tons.
the white hairs are a sign of cost reduction during cable production. they do put very little copper threads to save money and have to support the cable with cheap plastic threads in addition.
replace them with a proper cable which is thicker. you will hear the difference.
