Below is a short defense for who I believe wrote the book of Hebrews. I’ll be frank from the beginning and tell you that I whole-heartedly believe that it was Paul. Feel free to disagree, since I do recognize it is not a hill to die on, it doesn’t mean I shouldn’t aim to be a workman approved by God to accurately handle the Word (2 Tim 2:15).
There are several arguments for Pauline authorship. The first argument can be made from church history. Among the early fathers to recognize Pauline authorship is Origen (185/6-254), Eusebius of Caesarea (260/65-339/40), and Cyril of Jerusalem (386). The Muratorian canon (200) could also recognize Pauline authorship because it speaks of seven letters with authorial attestation. This leaves open the possibility that Paul could have written Hebrews without attestation. There were also others who held to Pauline authorship including Dionysius the Alexandrian bishop (199/200-265), the Antiochan synod of A.D. 264, and Clement of Alexandria recognized the community acceptance of Pauline’s authorship (160-215); Augustine also stated that the churches in the West included Hebrews with Paul’s letters. Homer A. Kent says, “In ancient times it was the belief that Paul was the author which caused its early acceptance in the Eastern portions of the church.”
A second argument for Pauline authorship is the content of the book itself. The content of the book parallels several of the themes that Paul addresses in his other unattested writings. Paul was in prison (2 Cor 11:23) and so was the author of Hebrews (10:34). Gromacki also cites the following doctrinal similarities: the preeminence of Christ (1:1-3; cf. Col 1:14-19); the authentication of apostles by divine gifts and miracles (2:3-4; cf. 1 Cor 12-14; 2 Cor 12:12); the humiliation of Christ (2:9-18; cf. Phil 2:5-11); the use of Israel’s wanderings as examples to contemporary believers (3:7-4:8; cf. 1 Cor 10:1-11); the temporary nature of the old covenant (8:1-13; cf. 2 Cor 3:6-18); and the emphasis of the faith principle (11:1-40; cf. Rom 1:17). Additionally, the OT quotation of Habakkuk 2:4 is quoted only three times in the NT (Heb 10:38; Rom 1:17; Gal 3:11); the other two instances outside of Hebrews are clearly Paul.
A third argument for the authorship of Paul is the closing of the book. There are several elements of the conclusion that could show Pauline authorship. First, the author calls God, “the God of peace” (13:20). This phrase is one commonly used of Paul in his other writings (cf. 1 Thess 5:23; Rom 15:33). Additionally, there is a reference to Timothy, Paul’s son in the faith, Hebrews 13:23, “Take notice that our brother Timothy has been released, with whom, if he comes soon, I will see you.” It is obvious that Paul’s relationship with Timothy was intimate (Acts 16:1-10; 2 Tim 2:1). In addition to this the phrase, “grace be with you all” is a common phrase used by Paul.
All in all, I would hold to a Pauline authorship of Hebrews because of the above arguments. While I understand that there are many other options and counter arguments to those listed above, these to me are the biggest arguments for the Pauline authorship of Hebrews.
 See also David Alan Black, “On the Pauline Authorship of Hebrews (Part I): Overlooked Affinities between Hebrews and Paul” Faith & Mission 16/2 (Spring 1999): 29-48; and David Alan Black, “On the Pauline Authorship of Hebrews (Part 2): The Eternal Evidence Reconsidered” Faith & Mission 19/3 (Summer 1999): 73-85.
 Eta Linnemann, “A Call for a Retrial in the Case of the Epistle to the Hebrews” Faith & Mission (Spring 2002), 21.
 Ibid., 21.
 Ibid., 21.
 Homer A. Kent Jr., The Epistle to the Hebrews: A Commentary (Winona Lake, IN: Baker Books, 1972), 17.
 Robert G. Gromacki, New Testament Survey (Grand Rapids: Baker Books, 1974), 320.
 Ibid., 320.