@Samhill I am aware of that tradition, but I have also seen discussion about the reverse - i.e. girls giving away buttons - as a manga trope, quite possibly as a joke related to that tradition.
Unfortunately, I can't find any reference to this, only to the tradition of boys giving buttons.
As I understand it, the idea is to give away a button that specifically fell off the girl's shirt because of her large breasts, as a way to tease or excite the receiver. I don't know if this is a real life tradition, but it does exist as a trope in manga. The only name I have seen for it is "paizuri", but as we have already established, that is not correct.
Anyway, I don't want to derail this thread any more than necessary, but if anyone knows the proper expression for the trope I'd be interested to learn it.