How did the children of Israel get their name?
They were descendants of Jacob who was a son of Isaac who was a son of Abraham (Exodus 19:3). Jacob, when concerned about his brother Esau's possible approach to kill him, stayed behind to pray all night. There he was visited by an angel. He clung on to the angel and refused to let him go until he was told that his request was answered:
"Then Jacob was left alone; and a man wrestled with him until the breaking of day. Now when He saw that He did not prevail against him, he touched the socket of his hip; and the socket of Jacob's hip was out of joint as He wrestled with him. And He said, "Let me go, for the day breaks." But he said, "I will not let you go unless you bless me!" So He said to him, "What is your name?" He said, "Jacob." And He said, "Your name shall no longer be called Jacob, but Israel; for you have struggled with God and with men, and have prevailed." Then Jacob asked, saying, "Tell me Your name, I pray." And He said, "Why is it that you ask about my name?" And He blessed him there. And Jacob called the name of the place Peniel: "For I have seen God face to face, and my life is preserved"" (Genesis 32:24-30).
The name "Israel" means "wrestles with God." It turns out to be a very appropriate name for his descendants because they constantly struggled against God for the entire existence of the nation as Stephen pointed out in his sermon in Acts 7.