The original figure came from Saturnalia, an ancient Roman holiday that took partying to a new level. Symbolism at that time dwelt on fertility. It could be crops, livestock, and children. The stag horns represented virility. They are still present on some costumes in various countries.
Similar to the mall Santas, teens don Krampus costumes the first week of December shaking rusty chains and terrifying small children. This custom still persists in some Germanic villages. Not too surprising when you consider the original Grimm Fairy Tales, not the sanitized Disney version, had plenty of horrors to scare children into appropriate behavior.
La Befana, the friendly witch, shows up with presents for good boys and girls. Naughty ones get lumps of coal. Befana is famous for her hospitality. When the Wise Men were looking for Baby Jesus, she invited them home and served them a meal. The Wise Men invited her to come join their search. She deferred, but changed her mind after their departure. She packed up toys her children had outgrown and set out after them.
She never caught up, which explains why she never got into the Nativity scene. Instead, she gave away the toys to children she met while traveling. La Befana returns year after year leaving gifts for children, and sometimes a piece of coal along with the other gifts as a reminder to improve behavior.
Instead of leaving out milk and cookies for la Befana, children place a bottle of wine and a salami for the beloved seasonal witch.