"To Grandma Negley
[This?] is surely our greatest comfort. Elsie Fairchild, 3 years old"
This is most likely the Elsie Fairchild who in 1910 lived in Benkelman, Nebraska with her parents Milton A and Kate H. That girl named Elsie was born in 1906, making this photograph possibly from 1909. In 1920, I believe I found Elsie and her parents in Waco, Nebraska about 250 miles northeast of Benkelman. Her mother was listed then as Helen Rose. I found a memorial at findagrave for Helen Fairchild in the Waco Cemetery, she died in 1925 when Elsie was 19. Then my search came literally to a dead end when I looked for her father. Milton is buried in the Mount Hope Cemetery in San Diego, California, along with Elsie. If I've found the right people, the three of them died in successive years, Helen in '25, Elsie in '26 and Milton in '27. Elsie has no descendants and Milton and Helen didn't have any more children.
"Grandma Negley" was not related to Elsie or her family in any way that I can determine. But this sweet little girl must have known my Great Great Grandma Sarah Negley somehow.