"Twist Daddy"? It's a binder post
, also known as a screw post
or Chicago screw
. She punched a hole in her plastic headband, pushed in the post portion, and secured it with... looks like felt. Then she made a button with the screw portion. Very creative, but ridiculously easy. I did these with my button machine, but you could probably use a Dritz-type button if you cut a hole in the pusher
The screw portion is available in a very wide variety of lengths.