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I was hoping someone would rise to the challenge and provide Eleanor with a double-dactyl, and sure enough, Shmuel sent along a nice one (thanks, Shmuel!):

Higgledy-piggledy
Eleanor Heffalump
Jumped from her box with no
Name on her head.

Owner Jed Hartman tried
Cyberdemocracy,
Spurned the results and chose
Nao's instead.

(Shmuel provided an alternate last line—"This one instead"—in case "Nao" was only one syllable, but in fact it's two; rhymes with "Day-o." Although back in the old days she used to sometimes make written puns as if it were pronounced like "now.")

Speaking of Nao, she notes that she didn't provide Eleanor's species. I have a query out to the person who I now think did; will provide credit when I know more.

Nao adds:

[The name] just popped out of the recesses of my brain, with no intermediary thought processes. Maybe Eleanor's telepathic?

Entirely possible.