Through the whole Bible: Genesis 17
Going through the whole Bible, you find that the darker chapters are frequently followed by chapters that are much more positive. Genesis 17 (Link) is one of those positive moments. Whereas Genesis 16 was not a happy chapter, showing the darkness of human nature, Genesis 17 puts Abram and Sarai back toward the right direction.
Abram and Sarai actually do not live through the end of the chapter. The opening aspect of this chapter is God declaring that Abram will now be known as Abraham and Sarai as Sarah. There's some valuable ideas that fall under this concept.
The first is this: naming rights are important. Think about it: how do major buildings get named? Or university properties? The names given are usually requested by the primary benefactor behind the project. Children are named by their parents. All of these processes of naming highlight this truth: the person that gives the name is usually either in charge or indispensible to the project. These days, it's typically that the person is indispensible for the funding, but that's not always the case.
By God changing Abram and Sarai to Abraham and Sarah, He is asserting their dependence on Him. It is a reminder that they are His and not the other way around: God is not named by anyone in Scripture. Abram has not decided what God to worship.
The second thing is this: names are symbolic of people. While Abram is the same body as Abraham, he's supposed to be becoming a different person from this point. Same with Sarah. From this point forward, they are to live up to the new name and to the calling God has given.
They are to consider their old selves as dead. The break with the past should be complete. Likewise, our break with the past should be total.
I am not saying that you should change your driver's license when you begin to follow Jesus. We should, however, be different than we were. As we abandon our old way, we should take on new ways and it should be obvious that we are different people.
Abram and Sarai and all of their old mistakes could not be destroyed, but they could be redeemed. Yet it took a change in them and their lives. That change was to be enough that they needed a new name.
We need that break from the past as well. Our new name is one we share with others around us: Christian. Let us live up to it.