Arduino hinge rotation sensor
Your answer pretty much depends on your precise mechanical arrangement.  If you can work it out such that a potentiometer can be turned as your hinge opens, that might be your easiest bet.  A rotary encoder, either absolute, or relative (with some mechanism to track where zero is) would be another fair option.

If there is no way to work a potentiometer or encoder in to your mechanical arrangement, then perhaps you can build  some mechanical arrangement where you can measure how much string gets pulled from a string potentiometer.

It short, your options are limitless, and really come down to the mechanical situation.  I think to get a meaningful answer, photos or drawings of the specific situation are necessary.