Israel, Arabic Isrāʾīl, officially State of Israel or Hebrew Medinat Yisraʾel, country in the Middle East, located at the eastern end of the Mediterranean Sea.
The name “The State of Israel” has much to do with linguistic pride, and little to do with political philosophy or government structure.
The word “republic” comes from the Latin res publica “thing of the people”, which in turn was coined as a translation of the Greek πολιτεία. The idea of a Republic is very much an idea of Western philosophy and government, bound up with the history and ideals and legacy of ancient Greece and Rome.
During the revival of Hebrew as a spoken language (it had been mostly a written and liturgical language for ~1800 years), Eliezer ben-Yehuda and other revivers made a conscious effort to find ancient Biblical Hebrew words or roots to convey modern concepts.* For example, in Hebrew soccer/football is called כדורגל kaduregel “ball [of] leg” rather than a direct phonetic borrowing of the English word, as is common in other languages (e.g. Spanish fútbol, French football, or Russian футбол).
Because biblical-era Israel was not part of the Graeco-Roman world by religion, philosophy, or other traditions (at least not until the Hellenistic Era), ancient Hebrew doesn’t really have a word that means “republic.” One of the ancient Hebrew words for a political entity that does exist in biblical texts is מְדִינַה (related to Arabic Medina). This word is typically translated into English as “state.”
The name chosen for modern Israel upon its Declaration of Independence was thus מְדִינַת יִשְׂרָאֵל medinat yisrael, “State [of] Israel.”
By the way, Israel’s founders also considered calling it אֶרֶץ יִשְׂרָאֵל eretz yisrael, “Land [of] Israel.” Most likely, they decided against this because the term is too linked to its usage in the Tanakh, and in particular to the idea that God promised the Israelites a region of land that is significantly larger than the modern state.
* Although it’s newer than this era, one of my favorite examples is the Hebrew word for computer, מחשב makhshev, which could roughly be translated as “thinkifier,” and is derived from the Semitic triliteral root חשב. (UPDATE: See Alon Shirizly’s comment where he points out that the use of this root to refer to calculation or computation is actually quite ancient.)