Baum doesn't say much about Aunt Em and Uncle Henry, in fact, he never gave them a last name, nor told us what "Em" is short for. (Emmaline or Emily?) In the MGM adaptation of The Wizard of Oz, Mrs. Gulch refers to Uncle Henry as "Gale," while in Disney's Return to Oz, Dr. Worley calls Aunt Em, "Mrs. Blue."
While Baum never said it outright, I think Dorothy's mother is Uncle Henry's sister. My only basis here is a sentence in The Emerald City of Oz, which begins "As for Uncle Henry, he thought his little niece merely a dreamer, as her dead mother had been." As it is Uncle Henry who seems to fondly recall Dorothy's mother, that is my only basis for claiming thus.
Other people feel that Dorothy is not a blood relative and was simply adopted by Aunt Em and Uncle Henry, and possibly adopted their last name, which works with Mrs. Gulch's statement in the MGM film. My problem with this is, how would Uncle Henry have known anything about Dorothy's mother?
The reason for Dorothy not having the same last name as her aunt and uncle is simply that if her mother married, she would have her father's last name, not her mother's. If she was Uncle Henry's blood niece, it would be impossible for them to share a last name without them changing it legally, or if Dorothy's father also happened to have the same last name, but was not part of the same family. (For example, my last name, Davis, is very common, but I'm not related to the guy who does the Garfield comic strip.)
I once came up with a nasty explanation of how Dorothy could have the same last name as her aunt and uncle and be Henry's blood niece: Dorothy was born as a result of adultery or rape, and for some reason, Dorothy's mother is no longer her guardian. Needless to say, no one, not even me, liked that idea when I shared it.
As for myself, I tend to ignore the names that have been given them by anyone but Baum.
I've been working on a screenplay of The Wonderful Wizard of Oz for awhile, and in writing, I decided it would help the story if I delved into the story of Aunt Em and Uncle Henry, even if it was mainly in my own imagination. I eventually built up a backstory for them, and Dorothy's parents. As I wound up with more story than could be incorporated in my screenplay, I started a fan fiction that I'm still working on, which told the story of Aunt Em through excerpts from her diaries.
In this, my own version, Aunt Em (I paid a tribute to the MGM film and called her Emily) has the maiden name of Stanton, which was the middle name of one of L. Frank Baum's sons, and was also used by Baum as a pen name. Henry has the last name of Carpenter, which was derived from an annotation in The Annotated Wizard of Oz (Chapter 1, Note 3), which tells of some in-laws of Baum's, who may have been the inspiration for the couple. The meeting I arranged for Emily and Henry was, in fact, based on how my grandparents on my mother's side first met. Except, while my grandfather served in World War II, I had Uncle Henry as a veteran of the Civil War.
A lot of the names I used in the story are derived from Baum's life and works, a few from my own family. (Marie, the middle name I give to Emily, was the name of my grandmother, and is one of the middle names of my sister.) I did throw in one reference to an Oz movie in the naming, Dr. Williamson is named after Nicol Williamson, who played Dr. Worley and the Nome King in Disney's Return to Oz. The names that Emily picks for her eventually miscarried baby are from main characters in Baum's Flying Girl and Aunt Jane's Nieces series.
I haven't gotten too far into the story, but in the latest part, Dorothy's mother and father have been married.
Dorothy's mother is named after Baum's mother-in-law, Matilda Jewell Gage, one of the late 19th Century's greatest feminists. Dorothy's father is named Charles, after one of Baum's favorite authors, Charles Dickens.
Eventually, Charles dies in the sinking of the USS Maine, the event that launched the United States into the Spanish-American War. Matilda and Dorothy go to live with Aunt Em and Uncle Henry, but something will happen to Matilda. Exactly what is going to be saved for the screenplay only, but it's no longer the snake bite that I once derived from The Diamondback, a short story by Baum about a rattlesnake.
What is important about Matilda dying in Kansas on the farm is not that I've worked out a nice way to say "how this happened," but the effect it will have on the family, most importantly Uncle Henry. In The Wonderful Wizard of Oz, Baum writes the following paragraph to describe Dorothy's uncle...
Uncle Henry never laughed. He worked hard from morning till night and did not know what joy was. He was gray also, from his long beard to his rough boots, and he looked stern and solemn, and rarely spoke.I took from this that Uncle Henry has closed himself emotionally. The reason, obviously, has not been disclosed by Baum, so I had to fill in my own: when Matilda died, it was in an event that Henry felt he could have prevented, and he feels that perhaps Dorothy resents him for it.
This would also have an effect on Aunt Em. With a niece who has just lost her mother, and a husband who refuses to speak, she is forced to be the strong one, who holds the family together. I once commented that if I was adapting the book to a modern-day setting, Aunt Em would be filing for divorce. She is partly relieved, in the final bit of "prologue" I set up for my script, when a neighbor who is moving away gives Dorothy a puppy, which helps her move on.
This leads up to a story that isn't told at all in The Wonderful Wizard of Oz, which tells what happened back in Kansas while Dorothy was in Oz. Uncle Henry, now realizing the loss of Dorothy, finally opens up to Aunt Em, and they reconcile at last.
The book The Wonderful Wizard of Oz ends with a reunion of Aunt Em and Dorothy, but I've always felt that the character of Uncle Henry is hurt by not being mentioned at all, and being almost snubbed by the story. If you search for "Uncle Henry" in the first book on The Ozmatron, he turns up 13 times, while "Aunt Em" yields 24 results. It's as if his character doesn't get the closure it needs, but Baum saved that for Ozma of Oz, which closes with Uncle Henry saying, "I'm better already, my darling," because Dorothy has returned safely from seemingly being lost at sea.
So, that is how I view Dorothy's aunt and uncle, two people with lives and stories behind them. You just got to look into it.