"Idiot" comes from the Greek word idios, meaning personal or private, a connotation that still survives in words such as "idiosyncrasy." So, originally, an idiot was simply a private individual -- maybe, but not necessarily, because he couldn't figure out how the doorknob worked and was trapped in his home.
Meanwhile, the origins of your mom being like a doorknob because everyone gets a turn are quite clear.
In ancient Greece, the term grew to be applied to the "common man" -- in other words, a private citizen, as opposed to someone holding a public office. (You read that right: In the good old days, there were no idiots in politics.) Of course, back then an ordinary person was almost always also an uneducated person. You see where this is headed. It wasn't long until the upper crust started extending the term from laypeople in general to "those stupid laypeople," and from there it was only a hop, skip, and a jump to Halfwit City, population: you.
And thus a whole history of snide class warfare worms its way into our schoolyard insults. It'd be like if you traveled to the year 3515 and saw doctors diagnosing low-IQ types as "NASCAR fan" or "flies coach."