As long as you connect the wires properly (correct pins on the potentiometer to the correct wires of the USB cable at both ends, correct pins on the Arduino) it should work properly.
USB extension cables are usually short enough that you shouldn't have any loss of your signal.
If you find the values "noisy" (vary a lot more than they did when the potetiometer was connected directly to the Arduino) then there's a couple of things you could do:
- Attach the ground (bare, braided wire) of the USB cable to the ground of the Arduino. This will shield the conductors within from (transmitted) noise from other circuits.
- Attach a capacitor from the wiper connection of the potentiometer to ground. A better way to say that is that you need a capacitor from the Arduino analog input to ground. Something like 100 nanofarad. This will short circuit noisy "spikes" to ground. A larger capacitor will make it smoother, but also slower to respond to real changes in the potentiometer setting.
You may find that the above doesn't do much for noise that occurs when you wiggle the wires.
Breadboards are notorious for poor connections, and most of your wires are stranded rather than solid as they should be to make good contact on a breadboard.
You could solder short pins of stiff wire to your stranded wires to make the breadboard connections better.
Better still, if you have a soldering iron just solder the connections and leave the breadboard out of it.
Be careful not to plug your cut USB cable into anything that expects a standard USB connection. You are misusing a standardized connector - there's no guarantee that other devices will "play nice" if they are connected to something they don't expect.
If it were me, I'd probably us the cable the other way around.
I'd use the end you normally plug into the PC on the potentiometer, and the other one on the Arduino.
The way you are going about it, you could potentially plug the Arduino into a USB host and "power it backwards." That could lead to damage of the host or the Arduino.
The way I suggest avoids that possibility.