Replacing a Piezo with a button
You don't really need to understand how the piezo works, because a button is much simpler.  Just take out everything connected to the piezo (R5 and D1) and replace it with a SPST, normally-open momentary button.  Connect one button pin to the GP0 pin and the other end to ground.

You'll have to change the firmware to enable the internal pullup on GP0, to use it as a digital instead of analog input, and to use some other source of randomness.  Maybe some sort of timer during debounce.