The question may be asking two separate things and this merits two separate answers.
1) Why use the name "Israel" at all and not "Palestine", which was the name of Mandate, "Judea" which was the name of the previous Jewish Kingdom, or any other name? -- Israel is representative of the Union of the 12 Tribes since all of those Tribes are the Tribes of Israel. Israel's mission was to bring all Jews together literally under one banner. So this name was chosen as a unifying Jewish term.
2) Why use the term "State of" instead of "Republic of" or "People's Democratic Republic of" or any other appellation? -- The Hebrew for "State of Israel" is Medinat Yisrael (מדינת ישראל) and the word "Medina" is ancient Biblical Word as opposed to Republic which is Republika (רפובליקה). Thus the State of Israel is harkening back to its Biblical Roots. Most terms that Israelis use for their government are Biblical Terms like Nasi (נשיא) which meant Prince, but now means President and Rosh Hamemshalah (ראש הממשלה) or Head of the Government is the Prime Minister.
The state of Israel was founded in 1948.
Although it was founded under the title of "The State of Israel", Israel is considered a modern country. For most countries, the word "state" means the same thing as "country". Think of Israel as a country with only 1 state in it. So it can be called either word.
It is the year when the state of Israel was founded.
Since the modern State of Israel was founded in May, 1948 .
The State of Israel is found in the same place it has always been, in the southwest Levant.
There was no U.S. state founded in 1948, but the state (or as we know it, country) Israel was born in 1948.
Citizens of the Modern State of Israel are called Israelis.Subjects of the Israelite monarchies were called Israelites.
The modern State of Israel was founded in the territory of the former British Mandate of Palestine in May 14, 1948.
It was founded on May 14, 1948 by David Ben-Gurion.
The state of Israel was founded in 1948. David Ben-Gurion was the first Prime Minister of Israel from 1949-53.
The Modern State of Israel was founded in 1948 which makes it 65 years old as of 2013.
The Ancient Kingdoms of Israel never abandoned the monarchic model until they were overthrown. The Modern State of Israel was founded as a democracy and never had a king.