We know that "once upon a time when Brahmadatta was reigning in Benares, the Bodhisatta came to life as a young hare and lived in a wood." (Jataka 316) He was then a very charitable bunny who offered even his own body. What is not mentioned then is that Sakka gave the hare some special powers by putting his image on the moon. This power the rabbit attained was a magical way of giving gifts. Those gifts materialised in the grass everywhere in the world in the form of colourful eggs. These egg-like presents are what one can still see on paintings like this one, right in front of the Buddha:
This is also the reason for the Easter Bunny to lay eggs, an ancient tale preserving the story of a former life of Shakyamuni. Such is the Buddhist origin of Easter.