May I suggest a slightly different approach? Rather than viewing the passage primarily as an historical record and thus exploring the jailor's original meaning behind his original words, I would suggest viewing this first and foremost as part of the story that Luke is telling and thus exploring Luke's usage of the word "saved" in the book of Acts. This methodology will allow you to get at the jailor's original meaning -- though not necessarily his original words -- through (what I would consider) sound interpretive methodology.
By saying this I am not in any way suggesting that the event was un-historical; rather, I am suggesting that the human authors of Scripture wrote intentionally, with specific purposes in mind, and they crafted their writings to these ends. So the author, Luke, used the language which best conveyed his intent to his readers. Luke's story most likely does not exactly reproduce the jailor's original verbiage.
To offer some support for this claim, consider NT "quotations" in Greek of OT passages which were originally in Hebrew. The "quote" is not the exact words used, but it does nevertheless accurately convey the meaning. Similarly, "quotes" from the Septuagint are not always verbatim. We see this also (I think) in the Gospels, where multiple authors record Jesus' words slightly differently. Each author's representation is faithful and accurate to Jesus' meaning, but it is not necessary for it to be a verbatim quote. (The idea that quotes need to be verbatim, in the original language, with [sic's] and all is a very modern standard. Keep in mind, too, that the quotation marks you sometimes see in your English translation were not there in the autographs.)
So in following this methodology, we need to first and foremost ask what is Luke communicating here? When viewed in this light, against the backdrop of his language throughout the book of Acts, it is clear that Luke meant this as a plea for spiritual salvation. It simply cannot mean anything else when approached from this angle.
The jailer wanted to get saved, made an inquiry to this effect, and the answer was no doubt "Jesus". What his exact words were -- or what language they were spoken in -- is irrelevant to the interpretation of this part of Luke's narrative.
"My understanding is that the jailor was likely a pagan Roman with little exposure to the Christian notion of salvation." I would imagine he had exposure to the Christian notion of salvation through Paul during his imprisonment.
"Was he referring to something within the framework of Roman mythology?" This is unlikely, given Luke's presentation of the inquiry and Paul & Silas' response.
"Or is this a reference to 'the way of salvation' proclaimed by the demon-possessed girl whose exorcism had landed them in jail?" In a manner of speaking, yes, as Luke presents them each independently as referring to spiritual salvation through Christ, though we cannot be sure that this was the jailor's point of reference.
"Or simply a request for rescue from the worldly dangers he faced?" This is unlikely, given Luke's presentation of the inquiry and Paul & Silas' response.
"Or is there reason to suspect that he had heard the preaching of Paul and Silas and was specifically asking for more information about their message?" This seems to be the most likely scenario. Additional evidence would be that Luke presents Paul as one sent by the Son of God to spread His message in new places, and the story of the jailer is presented as an example of Paul carrying out this work even amidst extreme difficulties.