I went to the experts. Good question!
Even in the earliest centuries the question as to the author of the Epistle to the Hebrews was much discussed and was variously answered. The most important points to be considered in answering the inquiry are the following:
(a) In the East the writing was unanimously regarded as a letter of St. Paul. Eusebius gives the earliest testimonies of the Church of Alexandria in reporting the words of a "blessed presbyter" (Pantaenus?), as well as those of Clement and Origen (Church History VI.14.2-4; Church History VI.25.11-14). Clement explains the contrast in language and style by saying that the Epistle was written originally in Hebrew and was then translated by Luke into Greek. Origen, on the other hand, distinguishes between the thoughts of the letter and the grammatical form; the former, according to the testimony of "the ancients" (oi archaioi andres), is from St. Paul; the latter is the work of an unknown writer, Clement of Rome according to some, Luke, or another pupil of the Apostle, according to others. In like manner the letter was regarded as Pauline by the various Churches of the East: Egypt, Palestine, Syria, Cappadocia, Mesopotamia, etc. (cf. the different testimonies in B.F. Westcott, "The Epistle to the Hebrews", London, 1906, pp. lxii-lxxii). It was not until after the appearance of Arius that the Pauline origin of the Epistle to the Hebrews was disputed by some Orientals and Greeks.
(b) In Western Europe the First Epistle of St. Clement to the Corinthians shows acquaintance with the text of the writing (chs. ix, xii, xvii, xxxvi, xlv), apparently also the "Pastor" of Hermas (Vis. II, iii, n.2; Sim. I, i sq.). Hippolytus and Irenaeus also knew the letter but they do not seem to have regarded it as a work of the Apostle (Eusebius, Church History III; Photius, Cod. 121, 232; St. Jerome, Illustrious Men 59). Eusebius also mentions the Roman presbyter Caius as an advocate of the opinion that the Epistle to the Hebrews was not the writing of the Apostle, and he adds that some other Romans, up to his own day, were also of the same opinion (Church History VI.20.3). In fact the letter is not found in the Muratorian Canon; St. Cyprian also mentions only seven letters of St. Paul to the Churches (De exhort. mart., xi), and Tertullian calls Barnabas the author (On Pudicity 20). Up to the fourth century the Pauline origin of the letter was regarded as doubtful by other Churches of Western Europe. As the reason for this Philastrius gives the misuse made of the letter by the Novatians (Haer., 89), and the doubts of the presbyter Caius seem likewise to have arisen from the attitude assumed towards the letter by the Montanists (Photius, Cod. 48; F. Kaulen, "Einleitung in die Hl. Schrift Alten und Neuen Testaments", 5th ed., Freiburg, 1905, III, 211).
After the fourth century these doubts as to the Apostolic origin of the Epistle to the Hebrews gradually became less marked in Western Europe. While the Council of Carthage of the year 397, in the wording of its decree, still made a distinction between Pauli Apostoli epistoloe tredecim (thirteen epistles of Paul the Apostle) and eiusdem ad Hebroeos una (one of his to the Hebrews) (H. Denzinger, "Enchiridion", 10th ed., Freiburg, 1908, n. 92, old n. 49), the Roman Synod of 382 under Pope Damasus enumerates without distinction epistoloe Pauli numero quatuordecim (epistles of Paul fourteen in number), including in this number the Epistle to the Hebrews (Denzinger, 10th ed., n. 84). In this form also the conviction of the Church later found permanent expression. Cardinal Cajetan (1529) and Erasmus were the first to revive the old doubts, while at the same time Luther and the other Reformers denied the Pauline origin of the letter.
(a) The content of the letter bears plainly the stamp of genuine Pauline ideas. In this regard it suffices to refer to the statements above concerning the doctrinal contents of the Epistle (see II).
(b) The language and style vary in many particulars from the grammatical form of the other letters of Paul, as in sufficiently shown above (see III).
(c) the distinctive characteristics of the Epistle (IV) favour more the opinion that the form in which it is cast is not the work of the author of the other Apostolic letters.
Most probable solution
From what has been said it follows that the most probable solution of the question as to the author is that up to the present time the opinion of Origen has not been superseded by a better one. It is, consequently, necessary to accept that in the Epistle to the Hebrews the actual author is to be distinguished from the writer. No valid reason has been produced against Paul as the originator of the ideas and the entire contents of the letter; the belief of the early Church held throughout with entire correctness to this Apostolic origin of the Epistle.
The writer, the one to whom the letter owes its form, had apparently been a pupil of the Apostle. It is not possible now, however, to settle his personality on account of the lack of any definite tradition and of any decisive proof in the letter itself. Ancient and modern writers mention various pupils of the Apostle, especially Luke, Clement of Rome, Apollo, lately also Priscilla and Aquila.
Circumstances of the composition
An examination both of the letter itself and of the earliest testimonies of tradition, in reference to the circumstances of its composition, leads to the following conclusions:
(1) The place of composition was Italy (13:24), and more precisely Rome (inscription at end of the Codex Alexandrinus), where Paul was during his first imprisonment (61-63).
(2) The date of its production should certainly be placed before the destruction of Jerusalem (70), and previous to the outbreak of the Jewish War (67), but after the death of James, Bishop of Jerusalem (62). According to ch. xiii, 19, 23, the Apostle was no longer a prisoner. The most probable date for its composition is, therefore, the second half of the year 63 or the beginning of 64, as Paul after his release from imprisonment probably soon undertook the missionary journey "as far as the boundaries of Western Europe" (St. Clement of Rome, "I Epistle to the Corinthians", v, n. 7), that is to Spain.
(3) The reason for its composition is probably to be found in the conditions existing in the Jewish Christian Church at Jerusalem. The faith of the Church might fall into great danger through continued persecution by the Jews, who had put James, the head of the community to a violent death. Precisely at this period the services in the temple were celebrated with great pomp, as under Albinus (62-64) the magnificent building was completed, while the Christian community had to struggle with extreme poverty. The national movement which began shortly before the outbreak of the last Jewish war would increase the danger. These circumstances might lead the Apostle to write the letter.
(4) The Apostle himself declares the aim of his writing to be the consolation and encouragement of the faithful (xiii, 22). The argument and context of the letter show that Paul wished especially to exhort to steadfastness in the Christian Faith and to warn against the danger of apostasy to the Mosaic worship.
The chief importance of the Epistle is in its content of theological teaching. It is, in complete agreement with the other letters of St. Paul, a glorious testimony to the faith of the Apostolic time; above all it testifies to the true Divinity of Jesus Christ, to His heavenly priesthood, and the atoning power of His death.
"You Can't MAKE this stuff up."
They cast their silver into the streets, and their gold is like an unclean thing. Their silver and gold are not able to deliver them in the day of the wrath of the Lord. They cannot satisfy their hunger or fill their stomachs with it. For it was the stumbling block of their iniquity.