The etymology of the word "Easter" comes from an ancient pagan goddess of the spring named Eostre, related to German Ostara. According to popular folklore, Eostre once saved a bird whose wings had frozen during the winter by turning it into a rabbit. Because the rabbit had once been a bird, it could still lay eggs, and that rabbit became the modern Easter Bunny.
Eggs, like rabbits and hares, are fertility symbols of extreme antiquity. Since birds lay eggs and rabbits and hares give birth to large litters in the early spring, these became symbols of the rising fertility of the earth at the Vernal Equinox.
Christian missionaries discovered that many of the tribes they were trying to convert already had spring festivals celebrating fertility. The rabbit was one of these celebrations' main symbols, so the missionaries simply co-opted it to fit their needs.
Pagan goddess + Egg laying bunny story + crafty missionaries = Modern Day Easter bunny! ...or is it "Eostre Bunny"?