They are a great example of a band that doesn't really want to explain their band name origin. I might even guess that the name My Morning Jacket has no real origin, hence the mystery.
One explanation found in several places is that Jim James was visiting his old student-hangout bar (Boot's Bar in Lexington) which had been razed by a fire. Amongst the charred remains, he came upon a jacket (or robe, I've seen both described) which had stitched "MMJ", which he took to mean "My Morning Jacket."
I also found the explanation that it is slang for when a man wakes up in the morning with an erection, but an erection with a condom on it.
One online post collects a number of origins:
Interviewer: Where did the name My Morning Jacket come from?
Jim James: I have no idea
"There are just some things that are meant for us and there are other things that are meant to be shared with the public. Our name and the name of the studio is for us; the music is for the public." - Jim James, Neumu interview, September 2003
Perhaps more believable in its simplicity is when Jim James told Magnet Magazine that the name is just something he remembers scrawling down in a middle-school notebook. Or when he said "I was just lying on my bed, writing songs in a notebook, and I put 'my morning jacket' at the top of the page, it doesn't mean anything. A lot of things happen for me that way - things pop into my head when I'm riding down the highway or at inopportune moments, like when I'm falling off to sleep." (The Age interview, March 2004)
Tennessee Fire (1999)
At Dawn (2001)
It Still Moves [Vinyl] (2003)
Evil Urges (2008)