The phrase "Children of Israel" refers to the offspring of the Biblical patriarch Jacob, who was renamed "Israel" after he triumphed in a mysterious nighttime struggle with an angel. The name "Israel" denotes "one who has struggled with God". Jacob/Israel had twelve sons who became the progenitors of the biblical Twelve tribes of Israel. Subsequently when their descendants multiplied during the exile in ancient Egypt the Bible (Exodus) refers to them as the Children of Israel.
In the midrash, a rabbinic Jewish genre of Biblical interpretation, Jacob has the status of the greatest of the three patriarchs, since only he produced a righteous family. Abraham had both Isaac and Ishmael, but the latter was expelled from Abraham's tent due to the negative influence he was having on Isaac. Isaac in turn had Jacob and Esau , the latter being a hunter and sold his birthright to his brother. It is only Jacob/Israel who has sons who, after all their internal struggles, emerge as a united family all loyal "Sons of Israel".
Based on the New Testament, Christianity claims that the Christians are the "new" "Israel" that took over the mantle and name as the "Children of Israel" since the Jews rejected Jesus. This view is called Supersessionism. Many European settlers in the New World saw themselves as the heirs of those ancient tribes, hence one finds that they named their children and many towns they settled in with names connected to the figures in the Bible. The rebirth of the Jewish state in Israel with the simultaneous in-gathering of many Jews from the Diaspora presents a major counter-argument to the view that the Jews have lost their title as the Children of Israel.
See also: Israelites, Jew, Chosen people