Example of a button: incidentally another fun place on da small web (the button is also a link).

The TL;DR is that it just sort of became the standard because of being an early standard, and then because they are kind of a nice size. Something I don’t think the article mentions (I’ll admit I started skimming around the end) is that when displayed for fun, web folk often like to stack or grid these together, so it makes sense that if you’re making a new button that you’d make it the same size, so that it would fit in with all the rest.