I just got back from listening to session 3. In it Narnion reads the line "it came to me" several times. This is a minor inflection issue, but I wonder if it might convey a slightly different and perhaps more interesting meaning if it were read with the emphasis on "me", as in "it came to ME". In this case the meaning is that of all the members of Thorin's company (and goblins) in the goblin tunnels, it was ME the ring came to, not Gandalf, not any of the dwarves, or any of the goblins, but ME, the hobbit. So of course it's mine.
This same meaning is discussed later in the book, like at the council of Elrond, where that consensus is that yes, the ring was meant for Bilbo to find, and then to pass on to Frodo.
This same meaning is discussed later in the book, like at the council of Elrond, where that consensus is that yes, the ring was meant for Bilbo to find, and then to pass on to Frodo.