so the children would take the coins to the Chanukkah light to check the value of the coins and the parents could tell them they cannot, for the light is assur b’hanah, opening a discussion as to the meaning of the light and chanukkah.
i believe other gifts have no mikor in tradition and are a result of imitating the goyish custom at the same time of year.
I heard בשם ר יוסף טנדלר זצ”ל that after the נס of Chanukah the children of most of klal Yisrael weren’t used to learning and had to be bribed with money to do so (and the fact that we can get them to learn, even if it is through bribery, is part of the simcha, and the fact that we’ll pay money for this shows them how choshuv it is).
I have no idea where he got this from, though.