What is it called when eeny meeny miney moe is done using a held staff?

Solution 1:

I don't believe it has a standard name; like many playground rituals, its precise name (and whether it even has a name) seems to depend on the particular locale where it is played. I've found it called "knobbing up" and "knobbies" and "bottle caps" and in comments both "stacking hands" (found by Jim in a book) and "hand-over-hand" (contributed by Yorik) have been suggested.

One possible alternative is getting the upper hand, or just Upper Hand. There is a theory, widely repeated on the internet and beyond, that this common idiom originates with the childhood game. One representative story:

Get the upper hand

Meaning Take a dominant position.
Origin Various suggestions have been made as to the origin of 'get the upper hand' (or 'take the upper hand'). Prominent amongst those is that the phrase originated in American playgrounds, in the way that children select sides for impromptu baseball games. The method is for one team captain to grab the bat at the bottom, then the other captain takes hold above the first's hand and they progress hand over hand along the bat until the top is reached - the one left holding the bat having the 'upper hand' and getting first choice of player for their team.

("The meaning and origin of the expression: Get the upper hand", The Phrase Finder, Phrases.org.uk)

I very much doubt that this origin is correct, given that the phrase itself goes back to the fifteenth century:

It is no nede that I telle you of one of the moost noble pryncys of good remembraūce Marcus Attilius first a labourer of the londes / and af∣tir electe consul and connestable of the batails of rome / whiche by many a tyme ouircame and had the vppirhande and victorye of the men of cartage mortall ennemyes to the Romaynes. (English translation of Cicero, 1481)

This predates American baseball by a few centuries, though some ball-and-stick games predated the quote above (see Wikipedia). It is thus possible that the phrase emerged from this method of decision-making, either in a game that used something like a bat or with any other stick- or staff-like thing, but the game-in-question wouldn't have been American sandlot baseball. In fact, at least one source claims that

Upper hand is a saying that began with a simple 15th-century game. It involved a staff or stick, and two or more contestants. . . . This practice was adopted in the 19th century on American playgrounds to determine which side would bat first in impromptu baseball games. (Andrew Thompson, Hair of the Dog to Paint the Town Red: The Curious Origins of Everyday Sayings and Fun Phrases, 2017)

Unfortunately, Thompson doesn't include any more details about the game, and whether it was actually called something like "upper hand".

More to the point for the original question, I can't find any evidence that the phrase is actually used nowadays by children for this activity. However, the (folk?) etymology is now widespread enough¹, and the phrase "fits" the activity so well, that you might well be able to use the expression for the activity and be generally understood. I was able to find at least one example of this usage in print:

Vintage baseball players Al "Rocky" Belbol, with the Brooklyn Eckfords of New York,left, and Jamie "Mouth" Ford, of the Athletic Base Ball Club of Philadelphia, try to get the upper hand to determine bating order during the 2014 Base-Ball Exhibition & Fair at the Navy Yard in Philadelphia. (Photo caption in AP article "Vintage baseball proves a hit with modern players", Times Free Press, September 13 2014. The photo shows two men playing the game described in the question.)

Depending on your audience and usage, you might want to include some context for clarity's sake (especially if you don't have an illustration). Something like:

They played "Upper Hand" to decide who would sleep on the couch, using Grandfather's cane in the absence of a baseball bat.

¹ In addition to the example quoted above, see, for example, "Upper Hand", American Culture Explained; Doug Lennox, Now You Know Baseball, 2010; Charles Hodgson, Carnal Knowledge: A Navel Gazer's Dictionary of Anatomy, Etymology, and Trivia, 2015; and so forth.