In "A Three Ring Circus", I discussed the text painting of the line "And of course Henry the horse dances the waltz". Observing the original poster that inspired Lennon to write the song in the first place, however, you can see that the horse's name was actually Zanthus, not Henry. "Zanthus" is a bit unwieldy to sing, so it is not surprising that Lennon took liberties on grounds of artistic license (rather similar to Paul McCartney changing the original "Hey Jules" to "Hey Jude"), but there are any number of two syllable names that Lennon could have chosen that would have worked just fine. In addition to being easier to sing, I see two reasons why Lennon might have chosen the name Henry over other equally easy to sing options: (1) "Henry the horse" features alliteration as both nouns start with the letter "h"; and more significantly (2) to avoid confusion between George Harrison and George Martin, the latter was sometimes referred to as Henry - Martin's middle name (Emerick, page 6). And since much of "Being for the Benefit of Mr. Kite" was Martin's doing (he played the organ in the break section, was responsible for the calliope collage coda, and is probably behind the ingenious three-part tonal structure), Lennon's selection of the name "Henry" to replace "Zanthus" seems like Lennon's way of acknowledging Martin's contributions to his song. I have never encountered any quotes from either Lennon or Martin to support this idea, but the facts do fit and they seem to fit too well to be coincidence.
Though I included a photo of the poster in "A Three Ring Circus", I just now found a better resolution picture (in which you can read the text much more easily) and have included it below. (Click to enlarge.)
Emerick, Geoff and Howard Massey. Here, There and Everywhere: My Life Recording the Music of the Beatles. Gotham Books, published by Penguin Group (USA) Inc., New York, NY, 2006.