I want to allow the player some freedom as to what their One Button is, and I'm wondering which way I should go with it.

Should I have them map their button at the start, and allow them to change it in options later, or should I just have any button be valid for the single function a button press will result in, telling them "Press any key to play"?

Please weigh in on this if you have the time.

As I take it from this topic (https://itch.io/jam/one-button-game/topic/811809/what-do-you-mean-exactly-by-1-button) both options are ok. So you can map multiple or all keyboard buttons to one action or make a “select your button” menu. It should be ok as long as the player doesn’t have to use more than one button during game play.