Japan's official name is 日本国.
日 means "sun," 本 "means where it comes from" and 国 means "country."
So, 日本国 means "country where the son rises." Japan has such an honorable meaning in its name that Japan made China, who saw Japan as a tributary state, furious when a Japanese diplomat submitted a letter to the Chinese emperor (Many people believe it's the case).
Actually the mention to the meaning of Japan didn't outrage Emperor Yang of Sui (The letter said "The Son of Heaven where the sun rises, to the Son of Heaven where the son sets, may good health be with you").
According to the majority of Japanese historians, what mattered was "The Son of Heaven." In the letter both the Japanese and Chinese leaders were called "The Son of Heaven." Since the Chinese emperor didn't consider the Japanese emperor equal to himself, the letter triggered the anger.
Let me return back to the main discussion.
Currently, Japanese people call the national name either Nippon or Nihon.
On June 30th in 2009, it was questioned during the Diet (national assembly).
A representative regarded both Nippon and Nihon as acceptable because both names are used daily.
Although "Nippon" seems to more formal than Nihon since it's used by the United Nations and also written on the national currency, the representative couldn't decide an unitary name because both are so prevalent.
I can see a glimpse of Japanese culture in the decision.
The ambiguity and tolerance, things I can't separate from the Japanese.
When we cheer up Japanese national teams, we use "Nippon" because it makes a good rhythm.
Which one of these names sounds the best to you; Japan, Nihon or Nippon?