Historically speaking, God didn’t choose the Israelites. The Israelites chose God. During the ages of the kings, before the exile, the Israelites were polytheistic. Throughout the books of Kings and Chronicles, there is a constant competition between those who chose to worship the one god Yahweh versus those who chose Baal, Ashteroth, and the other gods of that region. It’s clear that the people who wrote the Bible preferred Yahweh, but they were never able to completely wipe out the practice of worshiping other gods.
It wasn’t until the Babylonian exile, when the Israelite priests and scholars were forced out of Jerusalem, when the temple, which contained God’s holy presence, was burned and destroyed, that the Israelites came to a new, more dedicated conception of the one God.
But, that’s more of an historical story. The Biblical story of the Israelites might go back all the way to Adam.
Adam’s sons included Cain and Able. Cain killed Able, and Cain was cast out to become a “fugitive and a vagabond” (Genesis 4:12-14). However, Adam had another son, Seth. Adam had other sons and daughters (Genesis 5:4), but the Bible follows Seth. This began the line of patriarchs. Seth had sons and daughters, but the Bible only focuses on Seth’s son Enos (Genesis 5:6). Enos had sons and daughters, but the Bible only focuses on one of them to continue the line of patriarchs. This goes on for several generations, with each patriarch in turn producing sons and daughters, but the Bible only focusing on one. No reasons are given for focusing on one particular son instead of another. Eventually, this leads to Noah.
Noah rode out the flood with his wife and three sons, and their wives. Noah’s three sons were Shem, Ham, and Japheth. At one point, Noah became very drunk and was “uncovered within his tent” (Genesis 9:21). His son Ham, the “father of Canaan,” “saw the nakedness of his father” (Genesis 9:22). His other sons, Shem and Japheth, covered their father, walking backwards so as not to look at him.
When Noah woke up, he “knew what his younger son had done unto him” (Genesis 9:24). For this terrible crime (whatever it was), Noah said, “Cursed be Canaan; a servant of servants shall he be unto his brethren” (Genesis 9:25). For his other sons, Noah said, “Blessed be the LORD God of Shem; and Canaan shall be his servant. God shall enlarge Japheth, and he shall dwell in the tents of Shem; and Canaan shall be his servant” (Genesis 9:26-27). So, Shem became the favored son while Ham became the least-favored son. This becomes important later, when the Israelites, descendants of Shem, are ordered by God to destroy the Canaanites, the descendants of Ham.
The Bible continues the lineage of patriarchs, from Noah’s son Shem, to Shem’s son Arphaxad, and eventually to Abraham. Abraham’s father, the Patriarch Terah, lived in Mesopotamia. After Teran died, God said to Abraham,
“Get thee out of thy country, and from thy kindred, and from thy father’s house, unto a land that I will shew thee: And I will make of thee a great nation, and I will bless thee, and make thy name great; and thou shalt be a blessing: And I will bless them that bless thee, and curse him that curseth thee: and in thee shall all families of the earth be blessed.” (Genesis 12:1-3)
God didn’t say why. God just promised to make Abraham the father of a great nation. Later, God promised Abraham land: “he LORD made a covenant with Abram, saying, Unto thy seed have I given this land, from the river of Egypt unto the great river, the river Euphrates:” (Genesis 15:18). This was the land of Canaan, the inheritance of Ham, the son of Noah, who saw his father naked.
Abraham’s son Isaac was the father of Jacob, Abraham’s grandson. Jacob had a wrestling match with an angel (or some such being). This angel gave Jacob the name “Israel.” “And he said, Thy name shall be called no more Jacob, but Israel: for as a prince hast thou power with God and with men, and hast prevailed.” (Genesis 32:28). From that point on, Jacob was named “Israel,” and his twelve sons became the fathers of the twelve tribes of Israelites.
This is the point where the lineage of the biblical patriarchs ends. The Israelites are God’s “chosen people,” but choices had been made in every generation to favor one son over others. These choices are not explained.
If we look up on the Internet why God chose the Israelites, we will see a lot of explanations like God wanted the Israelites to be the light of the world, or God wanted the Israelites to be a step on the path towards the Messiah. But these are explanations after the fact. They do not explain why these specific people were called upon to do these things.