A key determinant in translating δικαιωσύνη θεοῦ (“the righteousness of God”) is the flow of the argument in the whole of Romans 3. Romans 3 begins with a question: what advantage has the Jew? This question follows logically from the previous material in that Paul has just finished indicting his fellow Jews right alongside the non-Jewish nations arguing that they properly and justly stand under the condemnation of God. So, if the Paul’s Jewish kinsmen are justly condemned along with the Gentiles, then the question is natural: what’s so special about being a Jew?
Paul’s answer is that the Jews are special in that they were made stewards of God’s self-revelation (3:2). The problem is that they did not faithfully steward that with which they were entrusted. They did not proclaim the name of God to the nations. This raises the question as to God’s own faithfulness. God has promised to bless all the nations of the world through Israel; yet if God is to be just, he must condemn Israel for her lawlessness. So, God finds himself in a catch-22: how will God be faithful to keep his promise to bless the world through Israel and still act in righteousness in condemning Israel for her unfaithfulness? What is God to do?
All this is to make the point that the central question of Romans 3 is whether or not God will act according to his righteousness. Paul asserts that God must be proved true, justified in his words, and prevail in his judging (3:4). But how exactly is he going to do that when the law silences the mouths of all and makes the whole world, Jew and Gentile, accountable to God?
If the question of Romans 3 is how God will be found righteous when he must both bless the world through Israel and simultaneously condemn Israel, then the answer to that question comes in Romans 3:21-26. God reveals his own righteousness (δικαιωσύνη θεοῦ) through Jesus. Jesus is both the faithful Israelite through whom the world will be blessed and the one who propitiates (ἱλαστήριον, 3:25) the just wrath of God that condemns sin. He does all this to demonstrate his own righteousness (δικαιοσύνης αὐτοῦ, 3:25) and to prove that he himself is righteous (δικαιοσύνης αὐτοῦ, 3:26) by showing himself to be both just (or righteous), in that he condemns sin, and justifier (or the one who makes righteous), in that he blesses the world through Jesus, the faithful Jew.
So, what is the righteousness of God? In Romans 3 it is that attribute whereby God always does what he ought to do. He always does what is right. He keeps his promise to Abraham to bless the world through Abraham’s descendant. He maintains his justice by condemning sin. And he does all this so that he may be justified in his words and prevail in his judging. He does it to reveal his own perfect righteous character. That’s the righteousness of God.