The Hebrews often referred to God as "Ha-Shem," which means simply, "The Name." God first reveals his name to Israel in the book of Exodus as Yahweh, which probably means something like, "The one who is," or "The one who will be/causes to be." God repeatedly said that everyone would know that he is Yahweh, both the Egyptians and the Israelites. He demonstrated just what that means when he delivered his people from bondage (Ex 6:6-7), demonstrated his superiority over the gods of Egypt (Ex 7:3-5), provided food in the wilderness (Ex 16:12) and came near in the tabernacle (Ex 29:46). Name is more than just a label, it says something about who a person is. It is not unlike many of the Native American tribes where men earned names that said something about who they were and their character. This is why our savior was named "Jesus," which in Hebrew means "deliverer," and this is also why he was called "Immanuel," which means, "God with us."
There is another way "name" is used in scripture. Often times, people, places, or things are "called by the name," which denotes belonging and ownership. So, when the temple Ark of the Covenant was called by the name of Yahweh (2 Sam 6:2), it denoted God's ownership. When the sanctuary was called by God's name, it denoted that it was his dwelling place (Jer 7:12). When the city was called by his name, it demonstrated that it was God's city (Dan 9:18). When a woman got married, she was called by her husbands name (Is 4:1), which denotes belonging. When the Israelites are called by God's name (2 Chr 7:14), it denoted belonging.
God had told Israel that they would be, "My own possession among all the peoples," and "a kingdom of priests," and "a holy nation" (Ex 19:5-6). They were called by God's name, and therefore they belonged to him. Using the same language, Peter reminds us that we are "a chosen race, a royal priesthood, a holy nation, a people for God's own possession" (1 Pet 2:9). As believers, we are called by the name of Christ.
This is the background behind Luke's phrase being baptized into or upon the "name" of Jesus (Acts 2:38; 10:48; 8:16; 19:5). The "name" of Jesus is used in a similar way as the "name" of the LORD in the Old Testament. We are God's possession and are not our own (1 Cor 6:19-20). We have been bought with a price. Part of that transaction took place when we were baptized "into the name" of Jesus Christ. After demonstrating our faith through repentance, confession, and baptism, we belong to Christ and are called by his name.
As the sanctuary was called by his name, now we are called by his name. As God dwelt in the sanctuary in Jerusalem, God now dwells in us. As God demonstrated the meaning of his name through his acts that demonstrated his character, God now demonstrates the meaning of his name through us who are called by his name and are his temple. What an incredible and humbling privilege and challenge!