I always got the impression in the Austin Powers movies that this was a joke name we were supposed to laugh at, but I have no idea what the reference is.
Is there even one or is it just a silly name with no real significance?
I took it to be a pun on the academic term “basic exposition.”
Exposition is a summary of plot that has occurred off stage or off screen, as reported onstage or onscreen by one of the characters. The character “M” routinely supplied exposition to James Bond in the 007 stories, briefing him on the latest villain and his activities.
So one on hand. it is an in-joke, but on the other hand, it’s not a funny joke?
Thanks for the help. 
It’s Austin Powers after all. 
I think it’s a joke meant to be more clever than funny.
Most jokes aren’t funny after you have them explained. But when I was in the movie the first time, and heard him introduce himself as “Basil Exposition,” and then proceed to give exposition of the plot, I laughed.