Searching for a cheap low-friction rotary encoder
"A precision of 1 deg would be good. I have some rotary enconders, which would be suitable for that job, but unfortunately they all have a lot of mechanical friction (or lock quite firmly after every step)."

Well, no, you don't have some suitable encoders. If your encoders have detents they have MUCH less resolution than you need.

You're looking for at least a 9-bit encoder (512 steps), since an 8-bit encoder will only have 256 steps per revolution, which is less than the 360 your 1 degree resolution requires. 

Unless you can find surplus units, or get lucky finding a salvageable encoder from a junked piece of equipment, you are out of luck looking for under 10 bucks. By at least a factor of 5.