I know he used the name as a pseudonym:
But why that name?
It is a pun.
The name “Richard” ultimately means “wealthy”. So, Poor Richard is like call a tall guy Shorty, or a skinny kid Porky.
The full name of Franklin’s alter ego was Richard Saunders.
Why “Richard Saunders”? Dunno; why did Laurence Sterne pretend to be someone called “Tristram Shandy”?
Dunno – but I’m seeing the movie tomorrow, I’ll let you know if they answer that question. 
It was also the latest in a long string of pseudonyms for Franklin that included Silence Dogood, Martha Careful, Celia Shortface, and Busy-Body. He used these names to conceal his own identity for several reasons, one of which was so he could pose a question for himself to answer. Franklin’s brother, James, had previously published Poor Robin’s Almanac. Also, Richard Saunders was the real name of an almanac writer in England in the 17th century. These notes taken from the book Benjamin Franklin, by Walter Isaacson. It’s quite possible that Richard Saunders was my ancestor on my mother’s side.
Richard was also an extremely common first name at the time, IIRC, so it was the rough equivalent of “John Q. Public” or “Average Joe.”