back of the napkin is a saying, you're at the bar having a drink with your friends, you've used the front of the napkin you wipe food off your mouth, then someone mentions some interesting problem and you turn your napkin over and scribble on it to explore the details.

A synonym for informal, "off the cuff", or "in the moment".

I think you missed the parent's critique. The saying is not 'back of the napkin', it's 'back of the envelope', or 'on a napkin'. No one uses an old wadded up napkin that had a drink on it - but plenty of ideas have come from, 'get me that napkin over there, I've got an idea' (a clean napkin...).

Exactly.

Front/back of an envelope makes sense, but not so much on a napkin.