Clearly understanding what Paul means by "justification" (δικαίωσις) is one of the most important tasks to interpreting his theology at large. It is a massively important concept in the letters to the Romans and to the Galatians, where the very truth and substance of his gospel is at stake (cf. Gal 1.6-9).
I think it would be a mistake to understand "justification" merely as a judicial declaration of righteousness or something like that. I think Paul's understanding of δικαίωσις and δικαιοσύνη involve much more, and that in fact they touch upon ontological themes and aspects as well.
Consider Paul's argumentation in ch. 3 of Galatians:
The only thing I want to learn from you is this: Did you receive the Spirit by doing the works of the law or by believing what you heard? Are you so foolish? Having started with the Spirit, are you now ending with the flesh? Did you experience so much for nothing?—if it really was for nothing. Well then, does God supply you with the Spirit and work miracles among you by your doing the works of the law, or by your believing what you heard? (3.2-5)
After a few emphatic statements that a person is not justified by the works of the law (2.16, 21), he goes on to mention the Holy Spirit. In these quoted verses from ch. 3, he argues a posteriori from the experience of the Galatian Christians, as though that should settle the matter. His argument is this: you have received the Spirit already, and you did that when you believed, not after having done any works of the Law; therefore justification is not by works of the law. The implicit premise is that there is a tight conceptual connection between justification, on the one hand, and the reception of the Spirit, on the other. Another implicit premise is that the telos of works of the law would be to receive the Spirit/justification. Since they received the Spirit, since they accomplished the goal of works of the law without performing them, there would be no point in going back to works according to the law. That is Paul's argument. This argument presumes that reception of the Spirit and justification are the same thing, or at the very least are closely related.
His continued argumentation in ch. 3 likewise speaks of the promise made to Abraham, a promise Paul interprets as the promise of the Holy Spirit (3.14). Justification is about receiving the Holy Spirit, it is about being united with the Holy Trinity and deified. That is what Paul says. His point is that you received this promise when you believed and when you were baptized (3.16, 26-7). Christ received the promised Holy Spirit (see, e.g., Mark 1.10; Acts 2.33), and so did the Galatians when they believed and were united to Christ through baptism into him. Their justification therefore is their unification with Christ and their receiving God the Holy Spirit; in a word, it is their deification.
For all these reasons, our understanding of justification must be understood much more broadly. It is not merely a declaration of righteousness or something of the sort. Paul's argumentation goes far beyond that and touches upon themes that have nothing explicitly to do with that. I am not denying that there isn't a judicial element involved in justification; perhaps there is. What I am affirming, however, is that justification for Paul -- at least in his letter to the Galatians -- involves an ontological transformation realized through the reception of the Holy Spirit. It is a transformation of identity that calls for a new life in light of what has happened (Gal 5.16; Rom 8.12-4).