I wanted to collect my thoughts on the destination of Hebrews.
1. The most intriguing detail in Hebrews in this regard occurs in 13:24--"those from Italy greet you." It is an amazingly thin comment to base anything on, but it seems to suggest one of two things: 1) the author is writing from Italy or 2) there are people from Italy with the author ("the Italians greet you").
Most think #2 is the more likely. If it were #1, then it is unlikely the author is in Rome, since then he would have probably said, "Those here in Rome greet you." The suggestion that the author is referring to Priscilla and Aquila, possibly with him in Ephesus, has been made. They were from Italy (cf. Acts 18:2). I think the preposition here (apo) tips it slightly toward the Italians being "away from" Italy but others disagree.
So the majority way of taking this phrase is that the author is writing to Italy and that some Christians from Italy are with him.
2. A second set of arguments have to do with the "afterlife" of Hebrews in the early church. Its earliest attestation is in Clement of Rome and the Shepherd of Hermas, both of which were written in Rome. Clement is traditionally dated to the late 90s. So Ignatius of Antioch does not mention it, nor do we have record of Papias from Asia mentioning it. It is mentioned in Rome by two people who did not completely agree with it.
However, Hebrews is quickly accepted as Scripture in the East. By Origen it is more or less "in." But it took longer in the West. The key issue seems to have been Pauline authorship. The West seems to have had a memory that the author wasn't Paul. Only when this issue passed did the West more or less rest with the idea of Hebrews as Scripture. They seem to have had a question mark about the book that lingered long after any specific memory of its provenance survived.
These dynamics are usually taken to support a Roman destination for Hebrews.
3. The internal evidence of Hebrews' situation fits Rome more than any other location we know in the early church. The audience has been Christian for some time (5:12). In a previous persecution, they lost property and stood by leaders put in prison (10:32-34). I think the overwhelmingly most natural way to read 13:7 is that their leaders died a martyr's death, probably in that previous persecution.
The only martyrdoms we know of in the church up to the year 70 are Stephen and James in Jerusalem, then Peter, Paul and a host of others in Rome under Nero. Jerusalem, despite being a long standing suggestion for Hebrews, really doesn't fit the letter at all. The temple is not mentioned, only the wilderness tabernacle. It's a thoroughly Greek document, which could fit a group in Jerusalem but probably not the type of group Hebrews is writing to.
Meanwhile, there are numerous persecution bumps in Rome. In 49 Christian Jews were kicked out by Claudius. In 64 Nero put a bunch to death. 70 could not have been pleasant with Vespasian and Titus parading conquered Jews about the city and then killing them. Although I am not in the majority in the US, I think it should be obvious that Hebrews 6:1-2 makes little sense if directed at a Jewish audience. The items listed there were not beginning Christian ideas for a Jew, only for a Gentile convert. This is an instance of obvious anachronisms fixed for Paul not making their way into Hebrews.
Similarly, Romans pretty much says it's primarily for Gentiles (1:13-15; 10:1-2; 11:13). It has an imaginary Jewish interlocutor (2:17), and no doubt there were some Jews in the audience. Again, it is only anachronistic thinking that would assume there would not be both "weak" and "strong" Gentiles in Romans 14. We've had almost forty years now to get over these blind spots.
So the Roman church was a primarily Gentile church, especially after Claudius kicked the Jews out. But these Gentiles were conservative, more like Jerusalem liked than Paul (cf. Ambrosiaster's indication of this in the 300s). In short, it fits neatly the likely audience of Hebrews.