Why is the word grace so much more prominent in Paul's writings?
Grace becomes prominent in Paul's writings because with Paul, the grace of God takes preeminence. Before Paul, grace was mentioned more as one of God's attributes. Paul's gospel magnifies grace, teaching that it is not license and that God's grace is greater than any person's sin.
Les explains that if you count how many times grace appears from Paul onward, you get something like a couple hundred. The reason is that beginning with Paul, it is the grace of God that takes preeminence. Before that, grace was just more or less mentioned as one of His attributes. Les also notes that when Barnabas saw the grace of God, few people even today understand God's grace. Grace is not license; Paul was falsely accused of preaching that because God's grace is great, people may sin as they wish. But grace means God's grace is greater than any person's sin, always greater. Gentiles who came out of immoral idolatry were saved by His grace, and Barnabas saw it, was glad, and exhorted them to cleave to their spiritual position.