For follow-up discussion and general commentary on the topic. Comments are sorted chronologically.
Paul's name is not mentioned as the writer of the book of Hebrews because to include his name may have meant a death sentence for him. The Hebrews (the Jews) would not be happy with the author of this book! I note that the author includes the name of Timothy at Heb. 3:23 - who was Paul's close companion. Surely therefore Paul wrote it. Who else had Paul's understanding of the OT and of the revelation of Jesus?
Whether Paul wrote it or someone else, the book was written to believers.
Hebrews 13:5 And be content with such things as ye have, for He has said, I will never leave you or forsake you.
The whole New Testament is for believers in Jesus to know we have eternal life. The book of Romans is our guarantee of eternal life. Romans 1:16 For I am not ashamed of the gospel of Christ, for it is the power of God unto salvation to everyone that believeth, to the Jew first, and also to the Greek. Faith alone.
The gospel of John salvation is by faith alone. John 6:47. John 3:16-18. John 6:37-40. John 12:32.
Pauls writings for the most part end with the book of Philemon.
Hebrews was written in AD 68. Most of Paul's writings were before that.
That does not mean Paul did not write it, it just means we do not know and it is really not relevant to know. The whole bible is for us, but not all to us.
We are living in the day of Grace. The GIFT of God is eternal life to whoever believes on Jesus. First John 5:13. Romans 3:26-28. For He is the just and the justifier of them that BELIEVE in Jesus. Since salvation is a free gift to all who just believe on HIs name, John 1:12, then we have nothing to boast in. You cannot boast in a free gift that we did nothing for except believe which is not work. Romans 4:5. Romans 11:6. Romans 6:23 For the penalty for sin is death, but the GIFT of God is eternal life.
John 3:16-18. First John 2:2. John 6:40. Hebrews just reveals more truth of the gospel. Our assurance of eternal life paid for by Jesus one blood sacrifice for the sins of the whole world.
We know the writer wrote it under the inspiration of the Holy Spirit. `