The word appears in the bible 129 times (not including four other cases where it means "rain cloud", but that's unrelated). The Even-Shoshan concordance says it means "head of a tribe, head of a congregation, ruler, etc." One thing I noticed from looking at the concordance was that it shows up in some biblical books, but not others.
Milgrom notes this in Excursus 1, "Some Political Institutions of Early Israel", in JPS Numbers. He defines nasi as "chieftan" or "clan leader", and sometimes the leader of the entire tribe. He then writes:
The term nasi' occurs over one hundred times in the Bible in a striking distribution. It clusters in the first four books of the Torah and in Joshua and again in Ezekiel and the postexilic books. It is totally absent from Deuteronomy, Judges Samuel and all the other prophets.
The antiquity of the term nasi' is corroborated by its occurrence only among those non-Israelite societies that are nomadic: Ishmaelites (Gen 17:20, 25:16) and Midianites (Num. 25:14)
Sarna, in his comment to Shemot 22:27 in JPS Exodus makes a similar comment:
Hebrew nasi' is the title given to the chief of a clan or tribe in the period before the monarchy.
And in his commentary on Bereshit 34:2, Sarna notes that Hamor, is called the nasi of the land, and not a king, as the Canaanite leaders usually are, because "the ruler of Shechem has dominion over rural - that is, tribal - territory as well as the urban center ... Such a complex situation does not permit absolute power. Indeed, Hamor does not act like a king."
So from all this we see that nasi refers to a leader who is not a king, and therefore is absent from many sections of the bible that focus on the monarchy. There are some biblical synonyms for nasi, such as nasich נסיך and nagid נגיד. It might seem, due to their similar sounds that nasi and nasich are related, but that doesn't appear to be the case. Nasich, while meaning "prince" in current Hebrew, derives from the root נסך, "to pour", and as Klein writes, originally meant "he upon whom the anointing oil was poured". Nagid (which today is used as "governor", as in "governor of the Bank of Israel") does not share a common root with nasi, but as Klein writes, they share a similar path of development. Here is his entry for the etymology of nasi:
Derived from נשא and literally meaning 'lifted up, exalted'. For sense development, compare nagid (=leader), from נגד, 'to be high'. According to G. Hoffman, nasi literally means 'speaker', and derives from נשא in the sense 'he lifted up (his word)'; compare nagid, which may have also meant originally 'speaker, spokesman'.
Later the term nasi was adopted to mean the head of the Sanhedrin, e.g. Rabbi Yehuda Ha-Nasi. This Encyclopedia Judaica entry describes how it was used even after the end of the Talmudic period:
The title nasi persisted for many centuries and in different lands throughout the Middle Ages, sometimes as the title of a defined head of a Jewish institution, sometimes as an honorific title only, given to important personages and to sons of illustrious families.
But why president? My theory is that there had always been a clear distinction between a king - מלך melech, and nasi. This certainly doesn't mean that the word nasi ever meant a democratically elected ruler before modern times - but in earlier periods, when the role of king had very significant implications (particularly in Jewish law), there needed to be an alternate term for such a leader. And since nasi had been used throughout the generations, it was a natural choice for a translation of "president", when that term began to be used.
Of course, I wouldn't need to resort to conjecture if I could find any source mentioning the coinage of the modern use. Unfortunately, I wasn't able to find any such source. As I did when I researched artzot habrit, I turned to the archives of the Historical Jewish Press. The earliest mentions of nasi there meaning "president" appear in the 1860s, describing Abraham Lincoln (there's an interesting Hebrew article here about how Lincoln was covered in the European Jewish press at the time). It's unclear to me if this means that the sense was coined then, or if it had been used earlier, but no one happened to be writing about presidents before that point.
This 1858 article in Hamagid, mentions James Buchanan, but calls him the ראש ממשלה Rosh Memshala of America, a term that today means "prime minister" and sar שר, "minister":
By 1861, the newspaper Hacarmel describes Lincoln as nasi, but puts the term "president" in parentheses to help the reader:
The parentheses are gone by this 1862 article in Hacarmel, although they call him ראש נשיא - rosh nasi:
And the earliest mention I found of "president" in Hebrew was from this 1853 book discussing Napoleon III, who had the title of "prince-president".
As mentioned in this Wikepedia entry, the term "president" has a history going back quite a while before being adopted by the founders of the United States. But those usages weren't so well known, so I can see why I wouldn't be able to find any references to them in Hebrew texts. However, certainly there must have been some awareness of US presidents before Lincoln, even if the Jews were not paying so much attention to the goings on in the States at the time (and their press wasn't as fully developed).
If any of you readers know of earlier uses of nasi or other Hebrew synonyms for "president", or any other sources that discuss the coinage of the term, I'd love to hear...