An excerpt from a lengthy Wikipedia article on the phrase has this to say:
"...The earliest known published versions in the English language date to 1855, one of which used the words eeny, meeny, moany, mite, and the other, hana, mana, mona, mike. Alternate versions have also appeared in several other European languages.
Many stories exist about the "real" meaning of the first line, although the most commonly accepted theory is that they are just nonsense syllables. Another theory posed by Christopher Knight and Robert Lomas in their book, The Hiram Key, suggests that the words are the first numbers in the counting system of the pre-Celtic Britons. The Lakeland Dialect Society more specifically suggests that this is just one of many versions of Celtic sheep-counting rhymes, but it observes that a large number of 19th century English publications exist in which these counting rhymes were more often used to amuse children than to count sheep.
Another possibility is that the British occupiers of India brought a doggerel version of an Indian children's rhyme used in the game of carom billiards: "ubi eni mana bou, baji neki baji thou, elim tilim latim gou."
The "Tiger" in the rhyme could be a reference to the Tippu Sultan known as the "Tiger of Mysore" for his ferocity; he enjoyed hunting tigers and incorporated images of them into his flag. His defeat in 1799 allowed the British to conquer India..."