“…Before we come to the task of dividing this epistle, it should be noted that before the Council of Nicaea, some doubted that this was one of Paul’s epistles for two reasons: first, because it does not follow the patters of the epistles. For there is no salutation and no name of the author. Secondly, it does not have the style of the others; indeed, it is more elegant. Furthermore, no other work of Scripture proceeds in such an orderly manner in the sequence of words and sentences as this one. Hence, they said that it was the work of Luke, the evangelist, or of Barnabas or Pope Clement. For he wrote to the Athenians according to this style. Nevertheless, the old doctors, especially Dionysius and certain others, accept the words of this epistle as being Paul’s testimony. Jerome, too, acknowledges it as Paul’s epistle.
To the first argument, therefore, one may respond that there are three reasons why Paul did not write his name: first, because he was not the apostle of the Jews but of the Gentiles: ‘He who wrought in Peter to the apostleship of the circumcision, wrought in me also among the Gentiles’ (Gal. 2:8); consequently, he made no mention of his apostleship at the beginning of this epistle, because he was unwilling to speak of it except to the Gentiles. Secondly, because his name was odious to the Jews, since he taught that the observance of the Law were no longer to be kept, as is clear from Acts (15:2). Consequently, he concealed his name, lest the salutary doctrine of this epistle go for naught. Thirdly, because he was a Jew: ‘They are Hebrews: so am I’ (2 Cor 11:22). And fellow countrymen find it hard to endure greatness in their own: ‘A prophet is not without honor, save in his own country and in his own house’ (Mt. 12:57).
To the second argument the answer might be given that the style is more elegant, because even though he knew many languages: ‘I speak with all your tongues’ (1 Cor. 14:18), he knew the Hebrew language better than the others, for it was his native tongue, the one in which he wrote this epistle. As a result, he could write more ornately in his own idiom than in some other language; hence, he says: ‘For though I be rude in speech, yet not in knowledge’ (2 Cor. 11:6). But Luke, who was a skillful writer, translated this ornate Hebrew into Greek.”
-St. Thomas Aquinas in his Commentary on the Epistle to the Hebrews.
I think the last part of Aquinas' argument is key. Most authorship debates concerning Hebrews assumes that the epistle was originally written in Greek, in which case they say it couldn't be St. Paul because his Greek wasn't that eloquent. Yet if it was originally written in Hebrew and later translated, then there is no problem at all with St. Paul being the author. And we know that St. Matthew wrote his Gospel in Hebrew because he was writing to a Jewish audience and then it was later translated into Greek. Why then could St. Paul not have done likewise?