Why did God name Israel Israel?

The name "Israel" first appears in the Bible as the name given by God to the patriarch Jacob after his struggle with the angel
Israel means Struggler of/with God

Afterwards Jacob's descendants are know as Bnei Yisrael - The Children of Israel
and the Land as Eretz Yisrael - The land of Israel

The Children of Israel (Bnei Yisrael) were divided into twelve tribes descended from the 12 sons of Jacob.

For several hundred years the Land of Israel (Eretz Yisrael) was organized into a confederacy of twelve tribes ruled by a series of Judges.

In 1000 BCE, a monarchy was established under Saul, and continued under King David and his son, Solomon. Upon Solomon's death the nation split into two kingdom, Israel (Mamlechet Yisrael)in the north, and Judah (Mamlechet Yehuda) in the south.

The Kingdom of Israel was conquered by the Assyrian ruler Shalmaneser V in the 8th century BCE. There is no commonly accepted historical record of the fate of those ten tribes, which are sometimes referred to as the Ten Lost Tribes of Israel.

The Kingdom of Judah was conquered by the Babylonian army in 587 BCE

After a few generations and with the conquest of Babylonia by the Persian Empire some of the exiles returned to their homeland. Under the Persians then under the Greeks, Jewish coins were minted as Yehud - Aramaic for Judah - coinage.

A revolt eventually led to the formation of an independent Jewish kingdom, known as the Hasmonaean Dynasty, which lasted from 165 BCE to 63BCE.

The Hasmonean Dynasty eventually disintegrated as a result of civil war. The people, who did not want to be governed by a king but by theocratic clergy, made appeals in this spirit to the Roman authorities. A Roman campaign of conquest and annexation, led by Pompey, soon followed in 63BCE.

Later, Herod the Great was appointed "King of the Jews" by the Roman Senate, supplanting the Hasmonean dynasty. Some of his offspring held various positions after him, known as the Herodian dynasty.

After the Census of Quirinius in 6CE, the Roman province of Judea was formed as a satellite of Roman Syria under the rule of a prefect until 41CE, then procurators after 44CE.

In 66CE, the Jews began to revolt against the Roman rulers of Judea. The revolt was defeated by the future Roman emperors Vespasian and Titus.

Jews continued to live in their land in significant numbers, the Kitos War of 115-117CE notwithstanding, until Julius Severus ravaged Judea while putting down the Bar Kochba revolt of 132-136CE. 985 villages were destroyed and most of the Jewish population of central Judea was essentially wiped out, killed, sold into slavery, or forced to flee.

In spite of the failure of the Bar Kochba revolt, Jews remained in the Land of Israel in significant numbers. The Jews who remained there went through numerous experiences and armed conflicts against consecutive occupiers of the Land.

The Modern State of Israel (Medinat Yisrael) was established in and named after the Land of Israel (Eretz Yisrael)

So we see:
Yisrael = Israel
Yehuda = Judah
Yehuda = Yehud = Judea
Yehudi = Jew