This general idea is known as "sortition", and actually has quite a long history. In the United States, and many countries, sortition still has a role–people are randomly selected for jury duty.

However, in ancient Athens, one could become a magistrate (~a small-town mayor in modern times) through random selection. There was a term limit of one year, and you were only allowed to serve once in a lifetime. You also had to go through a basic examination called the dokimasia to be deemed competent.

Several other civilizations have used this method over history, typically with the intention of preventing corruption. https://en.wikipedia.org/wiki/Sortition