Shortly after the Sholes & Glidden Type Writer was introduced in 1874, copyists set up shop, new territory for stenographers. The machine they used was the all-caps model that would shortly become the Remington Standard No. 1. Little is known of this early class of typists, but advertisements suggest they were bold entrepreneurs.
The New York Daily Herald, October 1, 1876 —
The Chicago Tribune, February 11, 1877 —
The Cincinatti Recorder, April 2, 1877 —
From Browne’s Phonographic Monthly, November 1877 —

Mary E. Hill may be the earliest typist known by name, but not much more is known about her. She placed advertisements in Browne’s Phonographic Monthly quite frequently, but what became of her is uncertain. A Mary E. Hill assisted Frank Overton write Nature Study (see book here).
Update (1/5/19) —
I overlooked Mrs. L.A. Cones, whose name is listed in the April 2, 1877 advertisement. So Hills and Cones are likely the earliest typists by name. Cones has an interesting history:
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I’m really loving your research, Mark. So much new information so well presented. Thanks.
Thank you. This research is a labor of love.
These really were pioneers!
Yes, and I wonder what method of typing they employed. They must have been sufficiently competent to offer their services.
Updated this post with additional information about another named typist. See end of post.