I don't intend to spend the rest of my blogging career talking about who wrote Hebrews, but I thought it was worth one more shot. Objections 'voiced' to my defence of Pauline authorship seem to be these:
1. Paul didn't say he wrote it, and if the had, he would have done. Answer: no authorship is claimed in the letter, but someone plainly wrote it!
2. Yes, but 2 Thessalonians 3:17 ('I, Paul, write this greeting in my own hand, which is the distinguishing mark in all my letters...') says that Paul did identify himself in all the letters he wrote. Answer: no, actually it doesn't. He says that he penned a final (not opening) greeting in his own hand. 2 Thessalonians is one of the few places where he draws attention to that (see 1 Cor. 16:21, Col. 4:18, Philemon 19;) and the final greeting in Hebrews may well be 13:22-25. He does not, for example, draw attention to his own handwriting in 2 Corinthians - but see 13:11ff).
3. Hebrews could have been written with apostolic approval - like the gospels of Luke or Mark - and 'why could Hebrews not have been written by Barnabas, or Apollos, say?' Well, it could. But Luke and Mark were included in the canon because they were written with apostolic approval; Hebrews is there because it was believed to be by Paul. Barnabas, Apollos, Priscilla etc; - they're just later inventions to make the (imaginary, in my mind) problem go away, as far as I'm aware.
4. How do I account for the canonicity of James or Jude? That's a good question; perhaps we can come to it some other time. BUT for now - note that it is only a peripheral question to this particular issue. If you're trying to show that apostolic approval is or isn't the criteria for acceptance, then 'How about James and Jude?' is directly relevant. Otherwise, not so much.
Now, here's another Big Gun:
Who is the letter from? What was clear to the original readers is not immediately clear to us, because the writer does not give his name. Paul is the most likely candidate. There are lots of similarities between his other letters and this one, both in style and content. Everything is centred on the person and work of Christ. In addition, the writer has a close and affectionate link with Timothy (13:23). What seems to clinch it, however, is the fact that the author's final sentence is, 'Grace be with you all' (13:24). Every one of Paul's letters ends with some sort of similar benediction. Closing prayers for grace are his unique signature tune (see 2 Thess. 3:18).
The Christian church has had a long history, and the centuries bear witness that most scholars have accepted Paul as the author of Hebrews. The reason he throws some people off the scent by some changes in style, yet plays his signature tune without signing his name, is probably something to do with the awful persecution going on at the time. In the Second World War many Allied broadcasts went out to continental Europe using varying wavelengths, but particular tunes were used in the programmes so that discerning listeners could identify their source and pick up hidden messages encoded within them.
In such circumstances those who do not recognise the tune, or who cannot crack the code, are left guessing. Guesses about who may have written Hebrews include Apollos, Aquila, Barnabas, Luke, Silas, Philip the deacon and Clement of Rome. The list is actually much longer than this, but we do not need to pay any attention to it. The fact is that if Hebrews had not been written by an apostle, or by someone writing under the supervision or influence of an apostle, the early church would never have accepted it as Scripture. But it accepted it with very little hesitation. Early Christians hummed tunes which many others have never learned.
(from 'I wish someone would explain Hebrews to me', by Stuart Olyott, Banner of Truth, page 3.)