Let me drop my guess on the author of the book of Hebrews.

But before I do that, I must say I’m greatly persuaded that it’s not Paul.
The author of Hebrews may carry much of Paul's revelations, but he doesn’t sound like Paul at all.

Back in my undergraduate days, I was part of a Bible study group, and I could often tell who wrote an outline just by reading it.
In the same way, I know Paul’s writing style and while the author of Hebrews was clearly influenced by Paul, it’s never Paul.

Paul may have had deep revelations, but the author of Hebrews had serious skill.
What amazes me most is how he consistently compresses broad Old Testament concepts into a few profound lines.

Many Christians might not fully appreciate the book of Hebrews because they don’t have much background in the Old Testament.
For instance, the author might simply quote, “I’ll be to him a Father, and he shall be to me a Son.”
That line was originally spoken concerning Solomon.
But the author expects you to reason through the scriptures and realize Solomon never fulfilled the prophecies tied to the sure mercies of David.
To fully grasp that one line, you'd need to explore multiple Old Testament books.

Paul would teach you to understand.
But the author of Hebrews teaches you to reason.
Also, Paul would never suggest that he received his message second-hand, as the author implies in Hebrews 2:3.

I don’t think any New Testament book quotes the Old Testament as much as Hebrews does. And in its quoting of the Old testament, it does not just quote, rather it invokes the whole context of the scripture it's quoting in the shortest possible way.
It’s a deep book very deep.

Now, back to my guess on the author.
Some say it was Timothy, but Hebrews 13:23 makes that unlikely.

There’s someone else in scripture who was famous for his eloquence.
In fact, Paul refers to him as "the brother who is praised in all the churches" in 2 Corinthians 8:18.

I believe Acts 18:28 is also about him how he mightily convinced the Jews, using the scriptures, that Jesus is the Christ.

And if you read Hebrews, you’ll agree with me: whoever wrote it mightily convinced the Jews, indeed.

So my guess is Apollos.