What does eenie meenie mean?

Eenie meenie has no special significance. It is like a nursery rhyme. For example, children play catch-up games together again. Using this nursery rhyme to choose who is arresting people is similar to drawing lots. In fact, the complete version of this nursery rhyme is:

Yi Ni, Minnie, Minnie, Moi,

Grab the tiger/monkey/baby's toes.

If he shouts to let him go,

Yi Ni, Minnie, Minnie, Moi.

My mother told me to choose the best one,

That's Y-O-U o-u.

Not because you're dirty,

Not because you're clean,

Just because you kissed a boy/girl at the back of the magazine.

Eenie meenie is also the name of a song by Justin Bieber, which has no special significance.