The Easter Bunny is widely believed to have its origins in German Lutheran story telling about the Easter Hare.
Unsurprisingly, therefore, the first English language telling of this seasonal visitor that I could find using the ID method on Google is an 1856 translation of German Stories. See Red Beard's Stories for Children: Translated from the German by Cousin Fannie. Boston. Phillips, Sampson and Co page. 7):
Then come away; with all your fears,
And take the pictures from the shears,
The ugly owl the Easter hare,
And long nosed Caspar standing there
Bonnie Taylor Blake directs us to the Oxford English Dictionary (OED), where the Easter Hare can be found in English in 1851.
Relying upon Google alone, you might be fooled (as was I) into believing that until now no one seem to know when the term hopped into the English language.
Searching among the 30+ million books in the Google Library with the ID method - to find the English language etymological origins of the Easter Bunny - gets us back down the rabbit hole as far as 1900. Its in:Child-garden : story, song, play. By the League of American Mothers (1900), Vol. 9. Chicago. Kindergarten Literature Co. (page 135):
The Easter Bunny - All rodent rights reserved (B)Public Domain
The Easter Bunny Bounds into English
Bonnie Taylor Blake veraciously directed me to earlier sources than can be found with Google: She found the Easter Bunny hopping around in the literature in 1891!
Goes to show - Google Library has not yet scanned the earliest sources of everything.
Surprisingly, on Google Books, the term "Easter Rabbit" does not get us back any earlier than "Bunny," But then I tried an alternative spelling, and with it I got back 10 years further down the rabbit hole with Google than Bonnie with the term "Easter Bunnie": I found him hiding inside a magazine from 1881.
The Easter Bunnie (B)Public Domain
The Easter "Bunnie"
'Cries from all sides resounded of the "Easter Bunnie! oh, let us try to catch him, and see if he has brought us any Easter eggs,"'
Source: Godey's Magazine (1881 - p. 471) . Is 1881 the etymological origin of the English language term Easter Bunnie? Bonnie could not get back further than 1881 when she went digging on that term. So, unless an older bunny - or bunnie - hops or otherwise lickerty-spits our way, that is currently the case. Godey's is a USA publication. being a 19th century "lady's magazine" published in Philadelphia.
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Nullius in Verba
Read Nullius in Verba to discover how to use Google to discover the earliest use of Words, terms and phrases