The title of this blog post is click bait. I don't know who wrote Hebrews, and neither do you. But I can hazard some guesses about the context in which it was written. Here are some highlights from said guessing.
1) It was likely written after c. 50. Why do I think this? It's the reference to Timothy in 13:23. Assuming that this is the same Timothy that we know elsewhere in the NT, and that reference to him is not spurious, and given that in Acts he is shown as first entering into Christian ministry after c. 48 (cf. 16:1), we can judge it more likely than not that the text was written after around 50ish (using 50 as a good, round, number). Is this a hard terminus post quem? No. The text could conceivably have been written earlier; there's nothing ruling it out. But it does seem to shift the balance of probabilities to sometime after the Jerusalem council of 48 (known to us via Acts 15).
2) It was written before 70. In 10:2, the author asks rhetorically why the temple sacrifices have not ceased. Such a question is only comprehensible if they had not ceased. But they did cease in 70. Therefore the rhetorical question only makes sense before 70.
3) It was written to Jerusalem. The text is entitled Pros Hebraious, to the Hebrews (Hebraious being the plural accusative of Hebraios). We don't know if that is original to the text or not, but it's known by this name from a very early period. In fact, AFAIK we don't have any positive evidence that it circulated absent that name. As such, we can hardly exclude the possibility that it preserves data relevant for reconstructing the situation in which the text was written. And interestingly enough, we know of another group of early Christians who were designated by plural forms of Hebraios: the Hebrews of the early Jerusalem church (cf. Acts 6:1ff). In historical investigation, there are two things that I find hard to accept: appeals to coincidences and appeals to leprechauns.
4) The text was written by someone associated with the Hellēnistai. Again turning to Acts 6, we see the Hebraioi contrasted with the Hellēnistai. Both groups were Jewish and Christian, and the distinction seems to be primarily linguistic: those whose first language was Hebrew or Aramaic, and those whose first language was Greek. If the text was written specifically to the Hebraioi, it seems probably that this was in distinction to Hellēnistai, which in turn makes most sense if the author him or her self most likely identified with the Hellēnistai.
5) It was not written from Rome. In Hebrews 13:23, the author sends greetings from "those from Italy." This has been seen as evidence that it was written from Italy, which becomes identified with Rome. This seems unlikely. The language of from-ness suggests that we are dealing with people who originated in but are not currently in Italy.
6) But Roman Christians were present when it was written, and were quite possibly involved in its production. The statement that those from Italy send greetings suggests that the author was in the presence of Christians from Italy. We can more specifically suggest a connection with Rome on the basis of the fact that 1 Clement (which originates from Rome) seems to be familiar with Hebrews. I would suggest that the most reasonable hypothesis is that one or more Roman Christians were involved in producing Hebrews. This person or these persons then returned to Rome with Hebrews. One or more such persons was later involved in the production of 1 Clement. That this person, or one of these persons, was Clement of Rome himself cannot be excluded.
7) The text was most likely written from Corinth. We know that early on Hebrews was associated with the Pauline corpus, which combined with the reference to Timothy suggests that it emerged from persons close to Paul. As such, we should be looking in the heartland of Pauline Christianity for its origin. That means Greece and southern Asia Minor. Again, we note 1 Clement, written to the Greek city Corinth, and containing at least allusions to Hebrews. Building upon what I argued above, this makes a great deal of sense if the Roman Christian(s) involved in producing both Hebrews and 1 Clement knew that the former had been written in Corinth and thus could count on it having a special sort of currency in that locale.
My best, educated, guess (which is better than an uneducated guess): Hebrews was written between 50 and 70, probably to Jerusalem, most likely from Corinth and certainly not from Rome, by someone previously associated with the Jerusalem church as one of the Hellēnistai, with the assistance of one or more Roman Christians who lately were involved in producing 1 Clement. Note that with regards to date, the most likely expansion of that range is downwards, as the argument for 50 as a terminus post quem is actually weaker for the argument for 70 as a terminus ante quem (although earlier than that range and the Corinthian provenance would be called in question, as we have really no evidence for Christians in Corinth before 50).
Incidentally, of individuals known by name the best candidate here is Barnabas: he was associated with the Jerusalem church; his Cypriot origin makes him more likely to have been grouped among the Hellēnistai than the Hebraioi; he moved in Pauline circles; and he had a connection with Corinth. But I am reluctant to assume that the text was written by someone otherwise known in the tradition. If it was, I am very much surprised that that identity was not preserved. Barnabas was prominent enough a first-generation Christian that I find it strange that no one remembered him to be the author of Hebrews. I think it more likely that the author was a Hellēnistai who has otherwise been forgotten.