But to all who did receive him, who believed in his name, he gave the right to become children of God,
Ignoring for a moment that they usually quote it without the rest of the sentence (or even the rest of the thought, as happens most often with Romans 3.23), the verse seems a powerful statement in some respects to the Reformed position.
Post-reformation decision theology states, "If you believe, then you're born again," and John 1.12 is often trotted out as a proof text. Both the Reformers and the Catholics would disagree, however—regeneration (the new birth) precedes belief.1 (Naturally the two groups go in a somewhat different direction thereafter.)
So today someone mentioned John 1.12 as a proof that the Calvinistic persuasion that one must be regenerated before believing was wrong.
But I think that if we look a bit more carefully, we'll find a somewhat different picture painted here:
11 He came to his own, and his own people did not receive him. 12 But to all who did receive him, who believed in his name, he gave the right to become children of God, 13 who were born, not of blood nor of the will of the flesh nor of the will of man, but of God.
Something should immediately bother us here: God's giving people who are born of God (verse 13) the right to become children of God (verse 12). Doesn't this seem a little obvious?
We might attempt to get around this by saying, "Well, everyone is God's child," which is sometimes done. Being a child of God is, in other words, basically just contingent on being physically born. But we know that this doesn't square with the rest of the verse, which gives us the idea that the birth isn't physical.
A perhaps better approach is to treat the phrase "children of God" as a title. We're still stuck with the obviousness of the statement, but we avoid any glaring issues with the spiritual nature of verse 13.
The key in this approach is verse 11, where we read that Jesus came to his own people and was rejected by them. His own people were Jews. Recall that the Israelites were always God's children, and they were considered his children by being born a Jew. (There were, of course, means of naturalization, e.g., circumcision.) They considered themselves children of God by virtue of their physical heritage. We see this in John 8:
39 They answered him, "Abraham is our father." Jesus said to them, "If you were Abraham's children, you would be doing what Abraham did, 40 but now you seek to kill me, a man who has told you the truth that I heard from God. This is not what Abraham did. 41 You are doing what your father did." They said to him, "We were not born of sexual immorality. We have one Father—even God."
The Jews, in other words, considered God to be their father because Abraham was their father. They excluded Gentiles because of genetics.
This, I think, is the message of John 1.12. Those who did receive Christ were given the same rights and standing as God's children that formerly belonged exclusively to the Jew. This was not based in physical birth, but in spiritual birth. Those who received Christ were born of God and were invested with the right that belonged to the Jews, who rejected Christ (v 11).
The verse does not say that if we believe, we're born again. It speaks to the unity of the body of Christ (cf Ga 3.28). It is clear that man is not born spiritually of his own will, but rather of God. There is no synergy in the new birth.
[1] This follows from the fall. Fallen man cannot in any way remedy his condition; the flesh cannot believe. (The works of the flesh are enumerated in Galatians 5, and they're all evil.) We are dead in sin (Ep 2) and must be made alive before we can believe. This spiritual birth is "being born again," or regenerated. It doesn't happen by believing—the spiritually dead don't believe. They just can't.