The  Lester Levy Sheet Music Collection  has a copy, described as "Original version by Laura R. Smith and J.S. Fearis.  Verse and revised arrangement by Adele Girard and Joe Marsala; [as performed by] Horace Heidt and his Musical Knights".  The notes describe it as "published in 1917".  Unfortunately, the image links are broken at present.  Adele Girard herself had this to say about the song  The Sweethearts of Swing: Adele Girard and Joe Marsala;  by Phillip D. Atteberry:

"Little Sir Echo was written by a couple of unknown composers (Laura R. Smith and J. S. Fearis) in the late Thirties.  Joe and I heard it, wrote a verse for it, rearranged it, and made it popular at the Hickory House.  I used to call it our 'kiss and make up' song because ordinarily I sang the opening line, 'Little Sir Echo, how do you do?' and Joe answered, 'Hello, hello, hello.'  But once in awhile, if we had been spatting, Joe wouldn't answer.  When that happened, I made sure we did the song again later, and yet again if necessary, until I hear his 'Hello, hello, hello.'  Then I knew things were all right."

I haven't found the full text, but the following may go part of the way (partly from the link Sorcha gave):

Little Sir Echo, how do you do?
Hello (hello) Hello (hello)
Little Sir Echo, we'll answer you,  [or, Little Sir Echo, I'm very blue]
Hello (hello) Hello (hello)
Hello (hello) Hello (hello)
Won't you come over and play? (and play)
You're a nice little fellow we know by your voice [or, You're a very nice person, I can tell by your voice]
But you're always so far away (away).

Malcolm