There are several occasions when God changed a person’s name, but why does He do that?
Abram to Abraham
God called Abram (before his name was changed to Abraham) out of paganism. We don’t know why God called Abram and not others. It simply says “the LORD said to Abram, “Go from your country and your kindred and your father’s house to the land that I will show you. And I will make of you a great nation, and I will bless you and make your name great, so that you will be a blessing” (Gen 12:1-2). It doesn’t say why God chose Abram over someone else, just as God doesn’t tell us why we were chosen and others were not. Clearly, it was not because of who we were, but because of Who God is, “So Abram went, as the LORD had told him” (Gen 12:4). There is no indication that Abram said, “Who are you Lord?” or that he hesitated or thought about it before going. It only says he went as the Lord commanded. God knew Abram believed God because he obeyed God, so “When Abram was ninety-nine years old the LORD appeared to Abram and said to him, “I am God Almighty; walk before me, and be blameless, that I may make my covenant between me and you, and may multiply you greatly” (Gen 17:1-2). That’s when God said “No longer shall your name be called Abram, but your name shall be Abraham, for I have made you the father of a multitude of nations” (Gen 17:5). God changed Abram’s name, which means “exalted father,” to Abraham, meaning “father of a multitude.” God changed Abraham’s name because of the changed role he would have in the future…the father of a multitude of nations. That’s why God changed his name to Abraham. 
Sarai to Sarah
Abraham’s wife Sarai also had a name change. We read the account in the same chapter where God changed Abram’s name to Abraham where “God said to Abraham, “As for Sarai your wife, you shall not call her name Sarai, but Sarah shall be her name. I will bless her, and moreover, I will give you a son by her. I will bless her, and she shall become nations; kings of peoples shall come from her” (Gen 17:15-16). Sarai’s name was changed to Sarah and for the same reason God changed Abram’s name. Sarai signifies “my princess,” as if her honor were confined to her own family, but Sarah signifies a “princess,” which means she is now a princess of many and not just those in her immediate family, so just as Abraham was now to be called the father of many nations, Sarah would be called a princess among many nations. God changed Sarai’s name due to her role in history. God changes names because names have meaning in the purposes and plans of God.
Paul and Peter
In one of the most remarkable conversion stories in the Bible, Saul had just received letters from the Jewish religious leaders to travel and purge the church from Judea. This basically gave him the authority to beat, imprison, and in some cases, stone believers to death. He had already consented to have Stephen stoned and killed (Acts 7:58). In Acts 9:1 we read that “Saul, still breathing threats and murder against the disciples of the Lord, went to the high priest and asked him for letters to the synagogues at Damascus, so that if he found any belonging to the Way, men or women, he might bring them bound to Jerusalem” (Acts 9:1-2). Saul was on a mission…a mission to destroy the church, but we know that Jesus said not even death can prevail against the church (Matt 16:18), so Jesus has an encounter with Saul on the Damascus Road and says to him, “Saul, Saul, why are you persecuting me” (Acts 9:4)? Saul must have been confused. How was he persecuting Jesus when he didn’t even know Him? Because the church is the Body of Christ and anyone who persecutes the church actually persecutes the Lord Jesus Christ. After Paul was commissioned to be the Apostle to the Gentiles, Saul’s name was changed from Saul (which means destroyer) to Paul (which means little or small). Saul, the destroyer of the church, now became Paul the small or the Apostle Paul; a man humbled by God a thorn in the flesh. Then there’s the case of the Apostle Peter whose name was Peter but then changed to Cephas. Jesus changed his name from Peter (petros for “small rock” or “stone”) to Cephas (which means “rock”). God changed Peter’s name to match what he would be in the church…a rock as far as stability is concerned.
Believers are told that when they enter the kingdom, God will give them a new name. What that name will be, we just don’t know, but we do know He will give us all new names. Perhaps like in the case of Abraham, Sarah, Paul, and Peter, our names will reflect our roles in the kingdom or our personality or character. God seems to call things or people by what they do or the role they serve, like with Isaiah who proclaimed salvation and his name means “The salvation of the Lord,” or Ezekiel who gave encouraging verses for Israel’s future, and his name means “God will strengthen.” Jesus’ and Joshua’s name mean the same thing: “God is salvation.” That’s appropriate because Jesus would save His people from their sins.
God has a new name for you and it’s ready to be revealed when Christ returns with the New Jerusalem. With God, as with people, names have meaning, so to the “The one who conquers, I will make him a pillar in the temple of my God. Never shall he go out of it, and I will write on him the name of my God, and the name of the city of my God, the new Jerusalem, which comes down from my God out of heaven, and my own new name” (Rev 3:12). What name you’ll be given may be closely tied to what you will be doing in the New Jerusalem or what you did while here on earth. The Apostle John wrote, “To the one who conquers I will give some of the hidden manna, and I will give him a white stone, with a new name written on the stone that no one knows except the one who receives it” (Rev 2:17). What that name will be for you and for me, I have no clue; I do now there’s a new name coming, and God never names something or someone without purpose and meaning. It’s strange how our own children’s name fit them, and these names were selected before they were born. For example, Andrew is our oldest. His name means “manly,” and he’s 6 foot 2, 210 pounds. God knew them by name, even before they were born. God knew what He was doing, even if we didn’t know at the time.
Here is some related reading for you: 5 Young People in the Bible that did Great Things 
Resource – Scripture quotations are from The Holy Bible, English Standard Version® (ESV®), Crossway Bibles. (2007). ESV: Study Bible: English standard version. Wheaton, Ill: Crossway Bibles. Used by permission. All rights reserved.