Perhaps you can add specifying / scope conditions since the term means different things to different people.

Modern stand-up comedy - the sort that boomed in the 80s, - is generally linked to dinner club and resort MCs who told standard jokes between performances by musicians and other entertainers and/or the vaudeville stage of the turn of the century, but it is markedly different in content and delivery. The lines are blurry between that vaudeville style any other form of comedic performance, which appear to be pre-historic or at least ancient in origin. Some very early historical records describe comedic performers as part of everyday life. Xenophon‘s Symposium of 360 BC or so describes Phillippos the Comedian in ways that sound enough like stand-up that it is quite striking.

I think what we generally have in mind when discussing stand-up comedy can only really be dated to the first comedy albums which both (a) allowed for a more conscious literary and stylized form of storytelling and (b) exposed jokes to a wide audience who could now tell what was new and what was old. Comedy albums in turn reined upon a stable of content-producers based in New York who had highly distinctive content that could connect with a wide range of consumers.

So it’s essential to specify scope conditions on this topic since the story varies dramatically based upon your focus. The narrowest focus would probably be author-performer comedic content stressing point of view, rigorously tested before live audiences and consolidated as a set, performed in clubs and similar venues and on television and in audio recordings. That scope of stand-up comedy would exclude people like Milton Berle and is a distinctive comedic art that developed in the 1950s in NYC as self-consciously individualistic comedy writer-performers broke from the pre-written joke content of the Catskills / later-day Vaudeville performers.