The pun on "new year's money" goes even further than mentioned.
Otoshidama = new year's money.
Dama for tama which is also the first kanji of tamago or egg. This part was explained.
But otoshi is also a conjugation of otosu, which means "to drop", which is why hiyoko flies up all the way to, well, drop an egg.
![Big grin :D :D](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)