Thanks to Mindi Woolman at Indiana Landmarks for suggesting this story from Chowhound:
Ilene V. Smith writes on the origin of the Easter Bunny
...a tradition thought to be German in origin. Germans had celebrated Ostara, the ancient goddess of spring, herself a symbol of renewal and fertility. According to legend, Ostara turned a wounded bird into a rabbit in order to save its life. The rabbit retained the bird's ability to lay eggs and, to thank Ostara, decorated and gifted her the eggs. The rabbit eventually became known as Osterhase — a folkloric figure Germans brought with them when they settled in the United States, particularly in Pennsylvania, during the 18th century.
The Osterhase tale was eventually woven into Easter celebrations. In either the late 19th or early 20th century, chocolatiers celebrated Osterhase by creating hollow chocolate stand-ins using newly developed molding techniques.
Full story including how Peeps likely got so popular!