What does Romans 2:24 mean?

Paul quotes from Isaiah 52:5 to drive home his point that those under the law "dishonor God by breaking the law" (Romans 2:23). When Isaiah wrote, the name—the reputation—of the God of Israel was sneered at by other nations because His people were being oppressed. In contrast, Paul now says God is dishonored by Israel's own actions. By breaking the law of Moses, they give God a bad name among the Gentiles. Seeing that the Jewish people break their own God-given law while looking down on others, the Gentiles respond by speaking blasphemously about Israel's God.

This highlights an important point which applies to believers in Christ, today. When those who claim to be Christians behave in un-Christ-like ways, it dishonors God. Non-believers see those sins and blame them on the faith. As Paul made clear earlier in this letter, that's not a valid excuse for rejecting the truth (Romans 1:18–20). However, those who claim the name of God need to be extremely careful about the kind of reputation we create in this world.

The point Paul is building up to is that nobody can keep the law of Moses perfectly (Romans 3:10). Everyone breaks it, somehow and somewhere, bringing dishonor to God. Everyone deserves God's judgment as a result of their sin. The law only helps to reveal sin, so salvation must come from another source. That source is faith in Christ (Romans 3:22–25).