Carolyn Wells, who compiled A Nonsense Anthology (1910), pointed out that this rhyme "is not strictly a nonsense verse, because it was invented and used for 'counting out,' and the arbitrary words simply take the place of the numbers 1, 2, 3, etc." Wells also points out that this rhyme has "… almost as many versions as there are nurseries."
Eena, meena, mona, mi,
Bassalona, bona, stri,
Hare, ware, frown, whack,
Halico balico, we, wi, we, wack.
The Oxford Dictionary of Nursery Rhymes (1951), lists a similar version to the one above that they say was popular in New York as early as 1815 (their source is Notes & Queries, 1855):
Hana, mana, mona, mike ;
Barcelona, bona, strike ;
Hare, ware, frown, venac ;
Harrico, warrico, we, wo, wac.
Thanks and Acknowledgements
Eena, Meena, Mona, Mi can be found in A Nonsense Anthology (1910) collected by Carolyn Wells.