What does Romans 9:25 mean?
Once again, Paul reaches back into the history of Israel, preserved in the Scriptures, to support his argument. He has been showing that "not all who are descended from Israel belong to Israel" (Romans 9:6), in addition to insisting that God has chosen to show His mercy to some who are not Israel: to Gentiles.Here he quotes from a passage (Hosea 2:23) originally written about God's commitment to one day restore the exiled northern ten tribes of Israel. Under inspiration of the Holy Spirit, Paul applies Hosea's text to Gentiles, as well as Jews. Peter appears to have more loosely done a similar thing in 1 Peter 2:10.
In Paul's context, to those who were once not God's people and were not "beloved" by Him, He has now declared to be His people. He calls them "beloved." In this way, God has exercised His right to show mercy on whom He will show mercy, including the Gentiles.