From Memory In Oral Traditions
(some line breaks added by me)
"More suble poetics also exist. For instance, the first line of Eenie Meenie contains vowels that progress from front to back (as in the fee, fi fo of fee, fi, fo, fum or in ee, eye, ee eye, oh of Old McDonald Had a Farm). Therefore meenie, miney mo sounds better than miney, meenie, mo, and the order is unlikely to change.
In addition, the first word, eenie, is part of the second word, meenie. This is a sophisticated poetic device used by Jakobson (1960) in relation to the political slogan I like Ike, used by Eisenhower in his presidental campaign. Together these first two words of the first line have four repetitions of the sound ee. It could even be argued that the /m/ sound is used because it is the easiest for a child to make.
Changing words without reducing the amount and quality of the poetic device is not easy. Changing words while maintaining both the meaning and the poetic devices is even more difficult. To be convinced of this, it is only neccessary to try to find, for each word in the two rhymes, a word that fits as well as the original.
Consider the first line of Eenie Meenie again. It has only sound pattern, not meaning, but sound pattern is enough. As noted, eenie is part of meenie, and there is a front-to-back vowel progression for the middle words. Meenie, miney and mo alliterate. Ennie, meenie and miney rhyme with a sound that repeats as the first vowel of eenie and meenie. Mo rhymes with toe and go. The remaining sound, /n/, repeats in the same location in three words. The whole line repeats as the last line, where the one sungle-syllable word, mo, coincides with the person who is chosen. This change from the two-syllable pattern adds to the closure of the piece (Smith, 1968) and emphasizes the end and the rhyme scheme by putting a stress on the last word. Thus there is not a phoneme or even a distictive feature that can change without breaking some pattern.
The whole line can change - for instance, to the Scottish version Eena, deena, dina, do, noted later in the chapter - but any single change reduces a sound-pattern repetition. The multiple constraints outlined in Chapter 5 are at work here, resulting in many sounds being overdetermined. Thus a change from the /o/ of mo would break the front-to-back vowel progression and the end-ryhme pattern, and a change from mo to a two-syllable word would take the stress off the rhyme sound"