Please correct any of this if I'm incorrect.
From what I have seen, back between 1986 and 1989, there were Super Mario Bros. cartoons broadcasted in Japan. Princess Toadstool was constantly called, as I observed, "Pīchi-hime,", or ピーチ姫, which sort of sounds like Peach. (One interesting note: one of the cartoons features Mario being born from a shooting star/peach, and it becomes his symbol.)
It wasn't until the very early '90s when Peach came up. From what I heard, when the game Yoshi's Safari was first sold in the United States, they published Princess Toadstool's name as Peach because it sounds similar to Pichi-Hime. Unfortunately, due to Yoshi's Safari not being popular in the US, they switched it back to Princess Toadstool until Super Mario RPG gave her the name again, and then it stuck.
Example of 1988 Japanese cartoon (sorry no subtitles)
[youtube]http://www.youtube.com/watch?v=7xPmfG1gO-E[/youtube]
Now I want to figure out why King Koopa is named Bowser...