Barnabus traveled extensively with Paul and was taught by him. they were so close that the Greeks thought they were Greek gods, Daman and Pythias. Barnabus was an educated Greek Jew, and the Epistle to the Hebrews, when studied, shows itself to be written in a cultivated Greek style.
As I said, it was something I read that argued very convincingly that Barnabas could not have written it, but I don’t recall the reasoning. Aside from that, I have always believed that Barnabas or someone very close to him was the most likely author. I believe others have suggested Luke, but the style seems totally different from his Gospel and Acts.