There's the usual theories on who it was: Paul, Barnabas, Apollos.
But an interesting one that I came across (I forget where I read this) is the suggestion that it was Timothy. Which is unusual, since Timothy is mentioned in the third-person at the end of the book. But here's the reasoning behind this theory:
1. There are many 'Pauline' figures of speech and concepts in the book. Timothy was a close companion of Paul, and was taught by him, so it would be natural for Timothy's style and language of teaching to reflect Paul's.
2. The author of Hebrews excludes himself as having directly received revelation from Jesus, instead including himself among second-generation Christians, who received teaching from those directly knew Jesus. Paul emphatically claimed to have received his revelation directly from Jesus, but Timothy was a second-generation Christian, so he fits this statement.
3. On a linguistic basis (the author of the essay claimed) there are several close similarities between Hebrews and 1 Peter. A reasonable connection to make might be that they had the same writer. Silvanus penned 1 Peter. And Silvanus (assuming it's the same individual) was a close traveling companion of both Paul and Timothy. In fact, Silvanus and Timothy traveled apart from Paul at least a few times. Conceivably, if Timothy was the author of Hebrews, Silvanus could have penned it.
4. Contextually, the author of Hebrews is very concerned with the superiority of the new Covenant to the old Covenant. This may be explained by the fact that Timothy was brought into the Church at the height of the circumcision controversy, which may provide a motive for why the author of Hebrews wanted to demonstrate that the old Covenant system (which would have included circumcision) had passed away.
5. The essay author presented the above, then went on to give a hypothesis for the closing remarks, where Timothy is mentioned in the third-person. In verse 13.19, it is Timothy who refers to himself as '... so that I may be restored to your sooner', indicating that Timothy was in prison at the time he wrote the book, which originally ended at verse 13.21, with the common closing doxology ('Jesus Christ, to whom be glory forever and ever'), closed with an 'amen'. Timothy thus sealed the scroll and sent it off to be read, and came to the hands of Paul.
In this hypothesis, verses 13.22-25 come from Paul's hand as his 'stamp of approval' for what Timothy had written. Verse 13.23, which says 'You should know that our brother Timothy has been released' is Paul commenting back on Timothy's imprisonment in verse 13.19, and so Paul providing an update for those to whom he is passing on Timothy's teaching. If just these four verses (13.22-25) belong to Paul, as opposed to the entire book, it explains how the writer can say 'I have written to you briefly'. Paul is writing a brief 'letter of recommendation', so to speak, for the authenticity of Timothy's teachings.
Like any of the other theories, much of this is circumstantial. Of course it can make sense, but just because it is internally consistent does not prove it is true. I simply thought it was interesting enough to provide for thought.