There we go. I was searching Robert Johnson songs, and couldn’t find which one I was thinking of. So that one is also an entry for 1936.
I found “Back Water Blues” by Big Bill Broonzy where he does refer to himself as “old Bill.” That seems to be from 1954.
In the Beatles’ cover of Carl Perkins’s “Honey Don’t,” Ringo (who has the lead vocal) calls out the guitar solos with “Rock on, George, one time for Ringo.”
Another example from Robert Johnson, maybe his best known:
I went to the crossroad, fell down on my knees
Asked the Lord above “Have mercy, now save poor Bob, if you please”
And, very early on, he recorded “Big Bill Blues.”
D’oh! My eyes just glossed over the “Bob”. First song I checked. 
Also 1936.
ETA: And “Big Bill Blues”, also 1936. Seems to be a popular year for self-referencing.
Heck, as I first read the thread and @pulykamell’s mention of blues singers, I also had Robert Johnson in mind, but couldn’t remember that specific line in “Kind Hearted Woman”. I thought of the line “when you call Mr. So-And-So’s name”, but didn’t remember the line where Johnson also referenced himself in that song.
On their album named “Bad Company,” to boot.
Black Sabbath did it three years earlier, with the song “Black Sabbath” on the album “Black Sabbath”.
Obviously not in the running for first, but Everybody Have Fun Tonight by Wang Chung in 1986 has the line:
Everybody have fun tonight
Everybody Wang Chung tonight
“R-A-M-O-N-E-S, R-A-M-O-N-E-S, Ramones!”
I was just listening to some Lightnin’ Hopkins the other day and noticed that he inserted himself into a lot of his songs. Still doesn’t go as far back as Robert Johnson though.
Wikipedia says it was recorded in 1927 (his first recording) as “Big Bill’s Blues.”
Didn’t Bo Diddley mention himself in Who Do You Love? Got rock n Roll music with the Bo Diddley beat.
Ironically the one that might be arguable. 
Bo Diddley’s first single was “Bo Diddley”. Yeah, he did it all the time.
Sweet, we have a new leader! Big Bill’s Blues (1927).
That’ll be hard to top. As an aside, I didn’t know that Broonzy recorded as early as 1927. I thought he started in the thirties.
Uh-huh, it was the Manfreds
Some of the “deepest” lyrics to emerge from the 60s.
Here’s a Youtube video that marks it as 1927:
The lyrics are quite different than the 1936 version I found. (Plus the instrumentation contains a piano in the 1936 version).
Wins the thread, I think, and pisses me off that I didn’t think of it. Correct me if I’m wrong but didn’t JSB also use the same four chords (B-flat, A, C, B-natural) in the coda of the “Wedge Fugue”?
More recently, in Frank Zappa’s song “Motherly Love” the lyrics name and are about nothing but the group themselves. Released in June '66 on the Freak Out album, three and a half months before the Monkees (and their theme song) debuted on TeeVee.
I know Cab Calloway name-dropped himself a few times, but he doesn’t go back before 1930. Earliest explicit one I found was “Calloway Boogie” from 1948