All that can presently (November 2023) be gleaned about Elsie Thompson from sheet music covers and by Internet searching is that she was presumably an American pianist who has been featured on just one work namely What Are You Waiting For Mary? (1923) words & music by Walter Donaldson, and published by Leo Feist, New York, USA
Her portrait on the left is taken from that same work.
She is also credited as the co-author, together with Moe Thompson - a relation perhaps, of Strolling (1920) published by Joe Morris Music Company, New York, USA. That not being enough, she also recorded, as a pianist accompanying Moe Thompson, Somehow, I'm Always to Blame (1923) words & music by Roy B. Carson, and published by F.J.A. Forster Music Publisher, Chicago, Illinois, USA.

Surfers are welcome to contribute any further information about this performer.