The original meaning of the δικ- word group was “that which was customary,” but was used to describe what was right in judicial cases. It was used in the sense of “judgement, lawsuit, trial, and penalty.” In the Greco-Roman world, the word was used for fairness in a court of law. One was “righteous” if one behaved in accordance with Roman Law. One is righteous in the Greco-Roman usage of the word.
Is Paul using the word as a Jewish writer might, in the light of the Hebrew Bible, or is Paul using the word the way a Greek or Roman might? The classic view of Paul is that he is developing a legal metaphor for salvation. Justification means that the believer is “declared righteous” legally in God’s court; legally he is made righteous. For example, according to Cranfield, there is “no doubt” that Paul means “to acquit” rather than moral transformation by this word group (Romans 1:95).
For many representatives of the New Perspective on Paul, however, this is a good example of a case where Paul’s Jewish background is important (for example, James Dunn, Romans 1:40). Paul does not necessarily want to evoke a Roman Court scene in the minds of his readers at all. What he wants them to hear in the word is the character of God in the Hebrew Bible as righteous and faithful.
This is far from an arcane argument among biblical scholars hoping to sell a few books. This verse is the main theme of Romans – God’s righteousness is being revealed from heaven in the Gospel of Jesus Christ. The rest of Romans is going to be an exposition of the righteousness of God. If the traditional view is correct, then the focus of the gospel is on our legal declaration of righteousness. If Dunn and others are correct, we might read this line as saying “God’s covenant loyalty and faithfulness is being revealed.” The Gospel is therefore about God’s character, the focus is on how he has acted in history to reveal his character.