You can see if you can get a servo pot ( a variable resistor designed to be able to rotate 360 degrees and go through the zero), use an absolute optical encoder ( used in machinery to give the position of a shaft) or use 4 linear hall sensors to detect a magnet on the bottom of a shaft by the varying magnetic fields.
The servo pot works and gives an output voltage almost directly in degrees, but has a limited life and needs to be waterproofed very well.
The absolute encoder is digital and very accurate, but will be expensive, and again needs very good waterproofing.
4 linear hall sensors are simple, cheap and rugged, but need you to do some maths either using a look up table or calculating the sines and cosines on the go to get the position information. With this you will probably find it easiest to build it, and record the values for every degree of rotation then use it to make a look up table for the device.