I've done it a number of times.
I got old and could no longer see well enough for anything tiny and up close.
 
Then I bought a pair of 8X magnifying glasses. Actually, if you are still youngish and can still see up close, you could probably use a pair of drugstore things off a rack, 5X for a few bucks.

But, had I known how helpful a pair of magnifying glasses were from the start, I would have had an easier time, each and every time.

You need a small, low power, pin point soldering iron tip, small guage solder and the ability to work with cables about like eight hairs untangled from a brush.
So, tiny needlenose and sharp blades are possibly helpful as well.

Just use your best small needlenose or heavy duty pointed tweezers, should work fine.
You will need some patience.

Nothing difficult about recabling a headset, except being able to work with tiny wires and tiny terminals.
Have some small heat shrink on hand to finish the job well. Be sure to put the heat shrink on the cable before you solder.

Nothing to it, really. Just understand the small scale in which you are working.
BTW, those cables are not as small as what you find when redoing your turntable, upgrading to silver all the way through.

Good Luck!  It's totally doable!

Sounds just like rewiring a tonearm on a turntable. Yes, patience! I’m ok with a soldering iron, but tiny wires and it hands are a big test. I’m 51 and never needed glasses until the past few yrs. I’ve got a magnifying light that I rely on for any small work. Sometime in the next few days I’ll get to it.

We had two cats, then down to one. Wife got suckered into a one eyed cat. She’s a handful, she def has some PTSD. Wife thought kittens would help her. Nope she hates them. Now all four of them follow me around. Two are jumpers and I’ve got claw marks all over my back from them ambushing me. Every day is an adventure.

Ha! My wife and I have always had cats, and I’ve always had headphones with replaceable cables.