Fat Blue also recently appeared in A Celebration of Me, Grover! A lot of minor characters, not just Fat Blue, have been appearing sparringly (and I mean really sparringly) ever since the format changed in 2002 (and even before then quite a few characters appeared less often). Some characters have returned recently, even if not on a regular basis.
I've read that Fat Blue is the name of the type of anything muppet that he's made from, but that's the name that he's most commonly known as. It's the name that he's usually called in official information on Sesame Street (such as Sesame Street Unpaved) but he has also had other names.
I once read on the old Muppet central message board in a post by Jerry Nelson that he's often called Customer in the scripts. There was once a skit where he reveals that his name is Mr. Johnson. There was also a skit where he was called Fred.
In A Celebration of Me, Grover!, there was a scene where Big Bird called him Mr. Blue, but in the credits he was called Fat Blue Customer.
I have come up with my own theory on why/ how he has had so many different names. My theory is that his name is Fred Johnson but his nickname is Fat Blue. When he was at the flyers club roost (in A Celebration of Me, Grover!) Big Bird probably overheard somebody call him Fat Blue and thought that was his real name and therefore called him Mr. Blue. Good theory, eh?