Sorry, I was teasing you a little. 8) I’ve also heard “falls”, but I believe the original lyrics use the words “stays” and “in”.
I think the whole preposition question is related to the fact that people sometimes only know most of the original words in lyrics, and not necessarily all of them. If somebody changes the word stays to falls, that person might keep all of the remaining words exactly the same (i.e., they might still use “in” because they remember that the word “in” is part of the original lyrics). Or they might also change the word “in” to “on” because that sounds more logical to them in combination with “falls”. I know it would sound more logical to me. After all, rain does fall on things, doesn’t it?
Using the prepostion “in” might be seen as meaning “within a certain area” (the plain) or I guess you might even be able to interpret it as “in the ground”. And my opinion is that in works better (more logically) with stays than with falls in this case.
I looked for the lyrics:
stlyrics.com/lyrics/myfairla … nspain.htm
So, in the song both falls and on are also used, but not in the line (sentence) you quoted. “On” is only used in response to a question about where the rain “falls”.