"Why, St. Paul did of course!"
This was how Fr. Swetnam started off our first class on the Epistle to the Hebrews. Why does he think that St. Paul wrote the epistle. Precisely for the reason that most contemporary scholars think he didn't---there is no name attached to it.
You see, St. Paul was commissioned to preach the Gospel to the Gentiles, while St. Peter was commissioned to preach to the Jews. Thus, with St. Paul's love of his fellow Jews, he wanted to write to them also before his time on earth was up. He left off his name because he was encroaching on the jurisdiction of Peter. This is something (the idea of episcopal jurisdiction) that is totally lost to non-Catholics. In the Catholic Church it has always been, from the time of the Apostles to today, the rule that bishops are not to teach within the jurisdiction of another bishop. So, out of respect for Peter, Paul wrote his epistle anonymously.
Another, clue that St. Paul is the author of the epistle is that at the end of Hebrews, he says that he is sending Timothy to them, who had just been released from captivity.
St. Paul wrote Hebrews, and he wrote it while under house arrest!