I've got an hypothesis about how the legend of La Befana developed. No, I haven't bothered to research it much (if you call reading 3 non-scholarly books not researching) because, well, I don't feel like it.
Even if I'm wrong...think about it...this is probably how legends have developed in human history.
La Befana is a witch riding on a broom. From what I can tell there is no exact representation of her so her appearance differs with the storyteller, except she is always an old warty woman with a crooked nose, riding her broom. I am going to assume I don't need to post a pic here since you've all seen Halloween witches. OK.
So basically one of the stories says the 3 wise men visited her in Italy while she was cleaning her house. They were on their way to see Baby Jesus (and to pray to him like they did in Talledaga Nights). They invited her to accompany them but she thought it was a waste of time so she said no.
Later she realized what a glorious opportunity she missed so she searched in vain to catch up to the group. She is still traveling the world but now she delivers gifts to kids instead of searching for the 3 wise men. No word on how she is still alive after all these years while all the other characters in the story are dead, even the Savior. OK.
I don't think I'm taking liberties when I say this is supposed to be symbolic. Prior to Christianity hitting Italy, healers were apparently older, wiser women. That was their religion. If you are familiar with the corno (Italian horn) or evil eye, those are remnants of the old religion.
When Christianity came the accusation was the old practitioners were evil magic doers. To make it seem less appealing, they were said to have warts, crooked noses and backs, wrinkled skin, wear drab colors, and to be really ugly. I suppose the apron is a reference to her being dirty while cleaning house.
Being a "witch" was dangerous for a time in Italy. Not only was it, well, not Christian, it was also a female-centered religion--not acceptable to the Christians whose beliefs and teachings were and are male-centered. The religion went underground. Women who were well-regarded healers were replaced by saints and madonnas with similar attributes to co-opt the locals into Christianity.
So the story of Befana is a story of conversion. She realized the error of her ways and chased after Jesus, but it was too late. Now she is doomed to wander Earth for eternity, searching but not finding Jesus.
There are few writings on the subject. Raven Grimassi seems to be the most well-known, albeit somewhat controversial among actual practitioners of "witchcraft", author of books about Italian folk religion. Here's a link to a few--an interesting glimpse into how legends are created to meet the needs of human beings.
Wikipedia has an interesting article about La Befana, almost as poorly substantiated as mine. ;)
The Befana comes by night
With her shoes all tattered and torn
She comes dressed in the Roman way
Long life to the Befana!