the OP said:
Is there anything special about the zener I should select besides 5.1V?
Power is usually checked.
Before getting into the rest of what your trying to do, Impeadances matter. So, if the CPU has an input Z of 10 M. adding a few K of series resistance doesn't doo much. Your joystick pot effectively changes the POTS output Z with position.
So, voltage dividers work "properly" when they are connected to an ideal voltage source (Z=0). So, an example. Suppose the input Z of th CPU is 10 M. I could add two resistors 500K || 500K (+5/GND), middle to bias that input at 2.5V with nothing connected, but I have lowered the effective input Z of the system.
Another caveot is that OP amps and such are not ideal and there is always a little current flowing in the inputs. It could even be Femtoamps. If you don't provide a resistance for that current to drop across, the voltage builds up at the inputs. It's called a bias return path.
So, all of those ideal OP amp circuits don't work right unless you pay attention to the output Z of what's driving it.
Some time ago, I did a really neat over-range indicator for +-10V for a piece of laboratory instrumentation. I compared to +-110 and then triggered a bi-color LED for like 1 second.