From Wikipedia:
One major theory about the origins of the rhyme is that it is descended from Old English or Celtic counting, as can be seen in the East Anglian Shepherd's count, "Ina, mina, tehra, methera" or the Cornish "Eena, mea, mona, mite".[1] The first American record of a similar rhyme is from about 1815, when children in New York are said to have repeated the rhyme:
Hana, man, mona, mike;
Barcelona, bona, strike;
Hare, ware, frown, vanac;
Harrico, warico, we wo, wac.[1]
The rhyme seems to have been unknown in England among collectors until the late nineteenth century, although it was found by Henry Bolton in the USA, Ireland and Scotland in the 1880s.[1] He also found a similar rhyme in German:
Ene, tene, mone, mei,
Pastor, lone, bone, strei,
Ene, fune, herke, berke,
Wer? Wie? Wo? Was?[1]
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.[3]