The country's name in English is "the State of Israel." "State" is a bit unusual- why not "Republic" or something like that? And were any names other than Israel ever given serious consideration? "Zion" seems kind of natural, given the Zionist idea that promoted the country's formation.
I suggest asking this in r/Judaism, some people there are experts in Hebrew and others who are experts in Israeli politics and history.
I'll handle why it called itself "Israel" first.
"Judea" was a popular contender for the name. If you go to "Jpress", you can look up the December 7, 1947 edition of "The Palestine Post" (today published as the "Jerusalem Post"), which mentions that an AP reporter asking people in Jerusalem found support for the name "Judea". Aharon Reuveni, a writer, claims he was the first to propose "State of Israel". As he notes, the name "Judea" was somewhat nixed by the fact that the proposed state didn't line up with "Judea" at all. He also suggested, in his own writing prior, why the other names would not work, which you can read there. These arguments, as far as I've seen, held sway with those who made the decision.
As for "State", rather than "Republic", the word "Republic" wasn't used much in Hebrew and had no real history in Hebrew as an antecedent. The word "State" (Medina) is used in biblical texts, as is the word "Country" (Eretz). See for example Esther, Chapter 1, which says:
וַיְהִי בִּימֵי אֲחַשְׁוֵרוֹשׁ הוּא אֲחַשְׁוֵרוֹשׁ הַמֹּלֵךְ מֵהֹדּוּ וְעַד כּוּשׁ שֶׁבַע וְעֶשְׂרִים וּמֵאָה מְדִינָה:
Translation:
Now it came to pass in the days of Ahasuerus -- he was the Ahasuerus who reigned from Hodu to Cush, one hundred twenty-seven provinces
There, the word is used as "provinces", but it's no less useful as a word for "state" in other places, including in Rashi's commentary on other portions where Rashi uses it to mean "country" or state.