“Banshees are from Ireland. They’re the souls of women who died after their children, and they steal children from bed in the middle of the night. Mothers can sit in your children’s room in the night to keep the Banshee away, though.”
Banshees are, essentially, the Irish folklore version of a witch. Both witches and Banshees are evil, supernatural women. This particular version of the Banshee belief has a few variations with the more popular versions of the belief. In this case, the Banshee is based around children, as the Banshee is the soul of a woman who outlived their children, whereas in other recorded beliefs, they are based around the idea of death; Banshees appear when people are dying. In this variation, though, they are given a far more directly malevolent objective: stealing children.
The informant is a descendant of Irish immigrants, and his parents would tell him stories of Irish folklore as a child. The versions of the legends he learned, then, could be the result of Irish folklore mixing with some more American-developed beliefs, such as mixing Irish Banshees and witches.