Authorship of Hebrews
Expansion of Pauline Authorship
The only overt clue as to the authorship is the reference to Timothy in Hebrews 13:23. This, in addition to the Eastern/Alexandrian tradition of Pauline authorship, led many to believe that Paul was the author. This is supported by significant uncial evidence that places Hebrews with other Pauline works (Ben Witherington III, Letters and Homilies For Jewish Christians (), 18). However, it's been noted that the style varies widely from the writings normally ascribed to Paul. However, the Eastern scholars Origen and Clement of Alexandria posited that it would have been more likely for one of Paul's associates to have written it since the ideas are Pauline, but the language is not (Ibid., Peter T. O'Brien, The Letter to the Hebrews (), 24).
deSilva also notes that the Greek used in Hebrews is "superior" to anything else that he had written. He believes that an abrupt shift in style is nothing more than an attempt to hold onto Pauline authorship (deSilva, 24). It is worth noting that the amount of time that Paul spent in Tarsus in his formative years would have lent itself to the capabilities to write in such "superior" Greek. It's not that Paul couldn't write in such a style, it's that he didn't.
The Western tradition does not hold Paul as the author. In fact, I Clement (end of first century) is the earliest quoted use of Hebrews, though some find echoes of Hebrews in Polycarp (Gareth Lee Cockrell, The Epistle to the Hebrews, (), 4), though Paul is not noted as the author. It was not until the fourth century that the West's attribution of Paul as the author surfaced (Ibid, 4).
David Allen Black attempts to harmonize the tension felt by Origen and Clement by suggesting that Paul could have dictated the writing to Luke (David Allen Black, Who Wrote Hebrews? (), 20). Since dictation standards in the Ancient Near East were not what we would hold as standard (the amanuensis was given latitude, especially if it was someone as respected and close to Paul as Luke), this mediating position may be preferable to others.
Apollos has also been suggested as a potential author as he fits many of the criteria of one who could write such a work. However, this suggestion is damaged by the lack of any attribution of authorship to him at all (Ibid, 9).
In the comments to an extant answer, someone mentioned that "liberal theologians" have suggested that "a woman" wrote it. This was, in fact, Adolf von Harnack way back in 1900 who suggested that Hebrews could have been penned by Priscilla. I'll have to find more evidence for Priscilla, but from what I recall it has to do with her prominence in Paul's ministry and that she was a fairly notable name in the church. Also I recall this perspective being supported by saying that she kept it anonymous so that it would be accepted by Roman Jewish Christians on merit and not rejected because it was written by a woman. This makes sense but is an argument from silence and so any actual evidence would necessarily hold more weight.
Additionally, I have found at least one justification for Priscilla as author to be rooted deeply in a sexist perspective ("what man would apologize for issuing commands? The author must've been a woman.") which is also somewhat flawed logic.
So, what does it matter?
Theologically it doesn't really matter who the author was, unless you want to argue that Paul somehow went supercessional somewhere in the middle of his ministry. Even if one accepts Pauline authorship, these references can be explained in context, setting, and provenance. I would argue that authorship would not influence the theological foundation or canonicity of the work of Hebrews.
Having said that, determining the author would provide interesting insight into genre, literary, and rhetorical studies. If the author was Paul, then the definition of what constitutes an "epistle" may need to be expanded (or restricted). It would also expand Pauline studies and, perhaps, provide insight into the textual backing for Paul's attributed assertions. Additionally, the New Perspective would also be significantly bolstered if Paul could be conclusively established as the author.
If Luke was the author/amanuensis of Hebrews, then the literary implications are also significant. What influence does it have on Luke-Acts? Are there echoes of Hebrews in Luke-Acts or vice versa?
The significance of authorship does not weigh heavily on the overall theology, but rather on the implications for biblical studies. These implications are the sorts of things that help us color our theology and vivify our engagement with the texts in question, but should not affect doctrine.