Note to the reader: Well, I'm way late. I've got excuses but they won't benefit you. As such, I've got apologies. I'm going ahead and writing/catching up so that it's here for future reference.
Growing up, one of the TV shows we watched fairly regularly was Dragnet. If you know the show, you're familiar with the line: "The names have been changed to protect the innocent." After all, you'd hate to be blamed for something you didn't do--so a shield of anonymity can be helpful.
In ancient cultures, though, name changes weren't typically to hide yourself. They were used to mark significant milestones in your life or to show a new allegiance. Which, of course, to the new king, was a significant milestone in your life.
And so we come to the name change for Abram. The Lord Almighty determines that it is time to mark a new beginning for Abram and so renames him Abraham. (An interesting note in one of the newer Greek New Testaments makes the opening syllable more aspirate, like "Ha-braham," which would sound more Middle Eastern.)
The meaning change, based on Study Bible notes, is basically a shift from "Exalted Father" to "Father of Many (or a Multitude)." The "Abr..." part is the "father" commonality. Taken this way, it's primarily a reminder from God that Abraham's heritage is more than just Isaac. He'll only live to see his grandsons reach the teenage boy phase, which means he may be tempted to take them out on his own. Or at least give up on feeding them.
What difference does this make? Well, God alone has the right to make the change and this is part of the narrative where God pushes Abram to make an even bigger step. In this chapter, God commands the covenant sign of circumcision. We won't hit the details of that here, but realize that this becomes a long-lasting part of the identity of the people of Abraham. And as you read the New Testament, you see it's significant in the life of the early church. (Check Galatians 6 for some thoughts on the matter.)
God shows His sovereignty in bringing Abraham and Sarah a child--and note that God also changes Sarai's name to Sarah, showing that God is her sovereign, directly, without intermediary--and reminds them both through the change of names. Then, a sign of the covenant was established, one that would be memorable and permanent.
What of us? Do we remember God's covenant? Do we understand the permanence of the One True God's promises and covenants?