Justification refers to God rendering a final verdict on the sinner. At the (future) final judgment, God will declare we are righteous, on the basis of the gracious gift of Jesus on the cross (in 3:20 the verb is future passive, although in the negative, no one will be justified by the Law). In other places, the same verb is in the perfect tense, looking back on the cross and its effects on the believer today. The verb in Romans 3 is in the present tense (present passive participle), we are being justified at the present time.
Justification is effected by God’s grace, as a gift. By definition, a gift is something given freely. If you try to pay for a gift, then it is no longer a gift and you run the risk of insulting the giver. By combining justification with grace (χάρις), Paul focuses attention on God as the one who bestows a gift on humans out of his gracious character. Grace is “God’s goodwill in action” often (but not always) in his gift of salvation (Kruse, Romans, 185).
Justification is also “through redemption in Christ Jesus.” The noun (ἀπολύτρωσις) is associated with paying a price in order to buy a slave or paying a ransom to win the freedom of a kidnapped person. There is an inscription dating before 100 B.C. which Moulton translated as “offering money for the ransom of other citizens, he showed himself gracious at every welcoming of those who from time to time safely returned.” (Sylloge Inscriptionum Graecarum15 325: τισὶν δὲ τῶν πολειτῶν ε[ἰς] λύτρα προτιθεὶς (sc. χρήματα) ἔδειξεν ἑαυτὸν πρὸς πᾶσαν ἀπάντησιν τῶν σωζομένων εὐομείλητον. See MM 554, Moulton, “Lexical Notes from the Papyri,” The Expositor VIII 1.3 (March 1911), 475-481).
In this case an offering of money was publically presented as an act of grace (a free gift) to redeem citizens who had been taken captive by “barbarian invaders.” We cannot know the motivation for this benefactor’s gracious act on behalf of his fellow citizens. Presumably he did this for his own glory since he had his gracious deed inscribed on a monument.
In this metaphor for salvation, God is like a gracious benefactor who paid for the ransom to gain the freedom of those enslaved to sin. All of humanity was in rebellion against God, in a sense “captured by the enemy.” Beginning in Romans 1:18 Paul described God’s wrath as deserved because humans have provoked God by their rebellion and hypocrisy, so that all people fall under God’s just wrath. Now in Romans 3:3, God acts on behalf of rebellious humanity and obtains their freedom from their real enemy.