'...if chance you call it'
In these essays I usually call it Providence (with a capital "P"). It's who or what chose Frodo that was not the Ring's maker - that appointed the task to him. The Power far greater than the Fellowship that needed to direct its actions. It's God's presence on each page of LotR even though He never appears. And it's one of the things that seems to have grown deeper as the story developed over the 16 years it took Tolkien to write it, one of the things that I'd say became more Catholic in the revision.
Although the title of this essay was spoken by Tom Bombadil (referring to his happening along at the right moment to save the hobbits), much of the development over time comes through Gandalf, which is fitting in that he not only plays the role of a prophet but is the only one of the main characters who would have had any direct knowledge of Eru. In an early version of the time at Rivendell following the Council, Frodo laments his 'importance,' and Gandalf replies, 'Quite accidental! Quite accidental, as I keep telling you... by which I mean, someone else might have been chosen and done as well...' Contrast this with the letter following the book's publication in which Tolkien says that Frodo had the ability to get the Ring farther than anyone else living in Middle-earth at the time.
Another interesting bit of development comes in the conversation between Gandalf and Frodo at Bag End in which Frodo asks why he was chosen. In an earlier version, Gandalf answers the question by explaining why Bilbo chose Frodo as his heir. In the book as published, the question has become deeper and Gandalf does not venture an answer. Although Gandalf couldn't tell Frodo why he had been chosen, he says very clearly that 'you have been chosen.' For Elrond, the fact that the task had been appointed to Frodo was directly related to the statement that 'If you do not find a way, no one will.' Frodo's calling was anything but accidental.
As Gandalf tells Frodo in the Bag End conversation as published, Bilbo was meant to find the Ring, meaning that Frodo was meant to have it. This idea also seems to have gained importance to Tolkien over the years. In The Hobbit, Gandalf's choice of Bilbo as the fourteenth member of the dwarves' company seems to be mainly a source of amusement to Gandalf, although that deepens even by the end of that book. Then, in "The Quest of Erebor," originally meant to be an appendix to LotR, we learn how Tolkien looked at those events years later. Gandalf gives a detailed account of his choice of Bilbo and the growth of his awareness that not only would a hobbit be a good addition to the group but that it was vital that this hobbit go on this journey. As often with such things, he didn't know at the time why this was important, but he was open enough to the movement of Providence to know that it was.
The action of Providence is not always simple. For example, after Pippin looks into the palantir, Gandalf says that 'You have been saved, and all your friends, too, mainly by good fortune, as it is called' (kind of similar to Bombadil's 'if chance you call it'), because Sauron was too eager. But then Gandalf says that Pippin's looking into the stone was actually, in itself 'strangely fortunate,' because it kept Gandalf from revealing himself to the Enemy, which would have been disastrous. So rather than Providence simply intervening so that the incident did not turn out 'as evilly as [it] might,' it actually became something positive.
This is such a basic premise in LotR that it can sometimes be passed over. Besides those already noted, some other occurrences (in chronological order) are:
Gildor's statement that 'In this meeting there may be more than chance, but the purpose is not clear to me...'
Sam's '...but I have something to do before the end.'
Elrond's statement at the Council: 'That is the purpose for which you are called hither. Called, I say, though I have not called you to me, strangers from distant lands. You have come and are here met, in the very nick of time, by chance as it may seem. Yet it is not so.' He goes on to say that it is 'so ordered' (interesting choice of words) that those gathered there must decide how to meet 'the peril of the world.'
-- And I'm sure there are more direct mentions of this that I've missed, as well as the many ways it's woven into the very fabric of the story.