the children's counting-out rhyme is first attested 1855 as eeny meeny moany mite; form eene meenee mainee mo is attested from 1923. Another variation is eeny meeny tipty te.
Examples from the Web for eeny
When you don't have that in mind when you're going solo, it's an eeny miney mo setup.
Why, they could eeny most walk ashore, come mornin, if she holds.A Singular Life|Elizabeth Stuart Phelps
"Wait until our Decembers tweak the top of your imperial nose off," said Eeny, shivering in anticipation.
There was one consolation for Grace and Eeny in their solitude.
The solemn beauty of the falling night tempted Kate to linger, while Eeny went on to the house.
She was in such a violent tremor—her voice was so agitated, that Eeny good-naturedly said no more.