While the name “Mother Goose” is now most closely associated with nursery rhymes, it was originally fairy tales that were attributed to the mysterious personage. While Charles Perrault’s 1695 book of fairy tales was properly called Histoires ou contes du temps passés, avec des moralités (literally “Tales and Stories of the Past with Morals”), it was more famously known by its subtitle Les Contes de ma Mère l’Oie, or “Tales of Mother Goose.”
The term had already existed in French for some time before that, having appeared in works from earlier in the seventeenth century. It was around 1765 that the first book of nursery rhymes credited to Mother Goose was published, and that more or less trumped her association with fairy tales, at least in English. I’ve seen quite a few Mother Goose collections that had the old woman riding a goose (or, more accurately, a gander), a reference to the rhyme in which she actually appears as a character.
So who was Mother Goose? According to Wikipedia, she doesn’t seem to have been anyone in particular, or at least not anyone who can be readily identified.
There are popular rumors that she was either a woman from Boston named Elizabeth Foster Goose or the wife of Robert II of France, but there’s no real evidence for either of these. The latter rumor is attributed to Katherine Elwes Thomas, an interpreter of nursery rhymes who came up with political explanations for a great many of them, but didn’t have any hard evidence to support her interpretations. If Mother Goose was ever a real person, we don’t know who she was. It’s most likely that she was just a representation of all country women who might have invented famous stories and rhymes. Mind you, anyone who’s read Robert Rankin’s The Hollow Chocolate Bunnies of the Apocalypse knows that the real author of the rhymes was Wheatley Porterman. Or possibly Sredna Sredna.
L. Frank Baum played on the idea of Mother Goose with his own book of Father Goose rhymes, most of which are pretty forgettable. The book was a hit for its time, however, and established Baum as a children’s author.
As far as I know, the only real link between Father Goose and Oz in Baum’s own work is that, when he shows up in The Tin Woodman of Oz, Captain Fyter sings a somewhat modified version of the Father Goose rhyme “The Soldier.” Donald Abbott wrote a book called Father Goose in Oz, and while I haven’t read it, I understand that it makes Father Goose an old man accompanied by a goose called Bilkins, named after another pre-Oz Baum character, Mrs. Bilkins from the Our Landlady columns. Fred Meyer also came up with a “Father Goose in Oz” scenario on one of his holiday cards, his version having the man receive a magic umbrella from an emperor, and using it to escape to Oz with a princess. I can recall a mention in one of Fred’s letters that Jack Snow was working on a story starring Father Goose, but I don’t know if any other evidence of this survives.