Question: What is the significance of name changes in the Bible? I thought of Peter.
Response: Extensive articles on the subject of biblical names can be found in e.g., Unger's Bible Dictionary and The Interpreter's Dictionary of the Bible (and, without a doubt, in almost any other major Bible dictionary: you might try Anchor Bible Dictionary, Hastings Bible Dictionary, etc. - they surely have similar treatments).
A couple of thoughts: Abraham (Abram), Sarah (Sarai), Israel (Jacob) and Paul (Saul) come to mind as the most famous name changes besides Peter. In each of these cases but Paul's, God is mentioned as responsible for the change, and in every case but Paul's, there is a deliberately expressed symbolism: Abram becomes Abraham, "father of a multitude", Sarai becomes Sarah, "princess", Jacob becomes Israel, "prince of God" (or also "he who struggles with God"), Simon becomes Peter, the "rock" (or more properly "stone"). The Hebrew etymologies here in particular are open to interpretation, but suffice it to say that in the most famous name change situations, God is making a point about the individual concerned (memorializing their spiritual accomplishments, their spiritual potential, and His blessing of them). Paul may fit into that category as well. We are not told why he changed His name, but Acts is written by his protege, Luke, and is noticeably without excessive flattery of Paul, leaving a number of personal issues and questions we have unanswered. In Greek, paulos means small, and it is certainly a mark of humility to go from a great Hebrew kingly name to a self-effacing gentile name (as Paul did become a servant to the gentiles on behalf of Christ).
It is also significant that all of us who have accepted Jesus and follow Him faithfully in this life will receive a new name in eternity as well (Rev.2:17). So one thing to get from the name changes of famous believers (and the fact that we shall all eventually follow suit) is that we are not citizens of the world that is seen, but of the one that is to come (Phil.3:20), and it is there that our true names, our everlasting names are written (Rev.21:12; 21:7; cf. Phil.4:4).
Please see also the following links:
What does the name "Christian" mean?
The "New Name" Believers will Receive in Eternity (in Coming Tribulation #5).
Yours in Christ,
Bob Luginbill