In The Magician’s Nephew, Andrew refers to his godmother, Mrs LeFay, who gave him the powder he used to make the rings. Other than this and a reference to her possessing "fairy blood", nothing else is known about her. Does anyone have any more answers within canon? If not, speculations within the spirit and plausibility of canon are accepted.
As noted in the question, The Magician's Nephew contains little information about Mrs Lefay. The name is probably inspired by Morgan Le Fay, the half-sister of King Arthur, but this should not be taken as necessarily implying kinship between them - "Le Fay" simply means "the fairy".
Uncle Andrew's talk to Digory gives most of the details we know:
Have you ever heard of old Mrs. Lefay?"
"Wasn't she a great-aunt or something?" said Digory.
"Not exactly," said Uncle Andrew. "She was my godmother. That's her, there, on the wall."
Digory looked and saw a faded photograph: it showed the face of an old woman in a bonnet... It was not at all a nice face, Digory thought
She was locked up in prison, and was released shortly before she died. Uncle Andrew gives the further information that she was part fairy:
my godmother was a very remarkable woman. The truth is, she was one of the last mortals in this country who had fairy blood in her...
"I bet she was a bad fairy," thought Digory
Digory clearly gets the impression that she was evil, but possibly she had a change of heart before she died, as she ordered Uncle Andrew to burn the box of magic dust.
Some more details are revealed from one of Lewis' notebooks which includes a long fragment of a hand-written first draft of The Magician's Nephew. When C.S. Lewis died, he left strict instructions to his brother Warnie that all his papers were to be burnt. The story goes that Walter Hooper arrived at The Kilns (Lewis' home) just in time to snatch some of the documents out of the bonfire, including the notebook which became known as The LeFay Fragment. According to a blog posting by Tenethia South:
In it, Digory has a special power to speak with trees and animals, but loses it when he is convinced by Polly to saw off one of a tree’s limbs in order to build a raft. Mrs. Lefay, his godmother, comes for a visit and seems to be sympathetic to his problem. Although we would all probably love to find out if this early Digory ever regained his ability to speak to trees, the fragment ends abruptly when Mrs. Lefay begins to give Digory directions to her home.
A commenter on the blog who read the Fragment reveals that:
His [Digory's] aunt said, “but if you observe me carefully, I’m sure you’ll see why I think she’s rude, unmannered, and slovenly.”
Mrs. Lefay turns out to be the “Fattest and shortest” woman Digory had ever seen (perhaps dwarven?). Her black dress is covered in yellowish dust that she claims is snuff perhaps a precursor to the dust in The Magician’s Nephew?). She gives a pretty good Mary Poppins impression until Aunt Gertrude leaves, when she then reveals that she knows about Digory’s loss, and hints that there might be redemption. In her words “You [Digory] look exactly what Adam must have looked like, five minutes after he’d been turned out of the Garden of Eden”
The Fragment is reproduced in full in Walter Hooper's Past Watchful Dragons: The Narnian Chronicles of C.S. Lewis (1979), available on free loan from the Internet Archive. I give below an image of two pages of the handwritten notebook, which were shown at the Bodleian Library as part of the "Magical Beasts" exhibition in 2013. If Lewis' handwriting defeats you, they can be read more easily on pages 50-51 of Hooper's book.
RE: Mrs.Le Fay: Fay has a Celtic etymology meaning Faerie arising from Arthurian Legend (Morgaine) Le Fay is an evil sorceress and King Arthur’s half-sister and Lewis would have had more than a passing familiarity with these.
Regarding the origin story a reference from Roger Lancelyn Green (a friend and fellow inkling) as reported in: The Land Of Narnia: Brian Sibley explores the world of C.S. Lewis, Brian Sibley, Harper Trophy New York N.Y. 1998 p26 (available at the internet archive. Lewis had completed The Lion, The Witch and the Wardrobe in 1949, but publication only took place the year after, and he had meanwhile started two more Narnia stories while awaiting publication. Regarding one of these Roger Lancelyn Green relates that Lewis read him the opening lines of a sequel about a boy called Digory, who has magical abilities (he can communicate with animals and plants), and lives with an aunt (Gertrude) and has a (Faerie) godmother called Mrs Le Fay (with a rabbit Coiny in her bag). Digory loses his powers while helping Polly a girl - his neighbour. Mrs. Le Fay (to help him regain his powers) instructs Digory on locating a furniture shop that sells birds and pictures and on reaching it he is further instructed to:
“Go in there and you will see…”
Sadly, what happens next will forever remain a mystery as C.S. Lewis apparently abandoned his efforts here and never completed this manuscript!
But all is not what it might seem, while Brian Sibley (above) makes no reference to a manuscript of this fragment C.S. Lewis Biographer Walter Hooper drew attention to just such a manuscript fragment which has been controversial as its authenticity has been called in to question by another C.S. Lewis Scholar Kathryn Ann Lindskoog. (more of this in Kathryn Ann Lindskoog Sleuthing C.S. Lewis: More Light in the Shadowlands, 2001 (a revision of 1994 edition) Macon, Ga, Mercer University Press p 111 accessible through the internet archive here.