Pretty sure it's a pulldown resistor. I was going by this set of instructions which also calls it a pulldown resistor: https://www.arduino.cc/en/Tutorial/Button

But what I meant was, in my design, if I don't have that pulldown resistor connected, then Pin 52 floats and the button line randomly gets input. If I have that pulldown resistor, then it doesn't get random input and everything works as expected.

This can occur if, for example, I don't have the Arduino board docked up to my custom shield/sandwich PCB. The pulldown resistor is on that PCB, so if I've got the Arduino plugged into the USB port of the computer "bare" then sometimes I see the reset/pair routine get triggered randomly.

Since you described your situation as having a "dangling wire" connected to Pin 52, and that you had the reset/pair routine trigger randomly, I was thinking that if you also didn't have pulldown resistor then you were simply running into the same thing.