Portuguese missionaries arrived in Japan at the end of the 16th century. In the course of learning Japanese, they created several grammars and dictionaries of Middle Japanese. The 1603–1604 dictionary Vocabvlario da Lingoa de Iapam contains two entries for Japan: nifon and iippon. The title of the dictionary (Vocabulary of the Language of Japan) illustrates that the Portuguese word for Japan was by that time Iapam.
Historically, Japanese /h/ has undergone a number of phonological changes. Originally *[p], this weakened into [ɸ] and eventually became the modern [h]. Note that modern /h/ is still pronounced [ɸ] when followed by /u/.
Middle Japanese nifon becomes Modern Japanese nihon via regular phonological changes.
Prior to modern styles of romanization, the Portuguese devised their own. In it, /zi/ is written as either ii or ji. In modern Hepburn style, iippon would be rendered as jippon. There are no historical phonological changes to take into account here.
Etymologically, jippon is similar to nippon in that it is an alternative reading of 日本. The initial kanji 日 may also be read as /ziti/ or /zitu/. Compounded with -fon (本), this regularly becomes jippon.
Unlike the nihon/nippon doublet, there is no evidence for a *jihon.