The reference to Ancient Greek is a matter of taste, I prefer new words to either have a meaning or to have no meaning, i.e. to be random. I find words with nonsense meaning irritating and more difficult to remember.
At least "fermion" is just a distorted form of Fermi, which does not suggest any specific meaning.
On the other hand "any on", has a meaning, but a nonsensical one, i.e. its suggests a reference to any one of the "ons", but there is no such thing as an "on", as that is just a grammatical case marker.
By the way, the "-on" in "anion" is not the same as the "-on" in proton. While the "-on" in proton comes from "-om", the "-on" in anion comes from "-ont", which corresponds to the English "-ing", the suffix of the active participle (anion = upwards-going).
It might have been a good joke initially, but it should not have persisted as a scientific name.
The termination "-on" in the Ancient Greek words "electron" and "proton" does not have any concrete meaning, it is just the thematic vowel "o" followed by the marker "n", of a singular neuter noun (originally "m", but in Ancient Greek all final "m" became "n").
Combining some undeclinable English word with an Ancient Greek case termination seems ugly and pointless for me (not that "fermion" and "boson" are any better from this point of view).
Even if we accept making words to rhyme as a technique for coining new words, "anyon" remains a bad choice, because it is easily confused with "anion" (introduced by Faraday almost two hundred years ago).
I assume that "anyon" is meant to have a different pronunciation, i.e. like "any-on", but nonetheless, in almost all other languages the vowels are pronounced different than in English and the default pronunciation for "anyon" and "anion" will likely be the same.