What caused Frodo to finally give in to the power of the Ring and claim it?

[Mithadan]

I agree that it is very difficult to determine what aspect of Frodo's desires or character the Ring influenced in causing him to ultimately claim it. Frodo is in many ways an "enigma" (ooo that word!). As a Hobbit, he is atypical. He is well educated, unafraid of "adventure", purposeful and single-minded. In many ways he is far more noble than any other Hobbit we see. Merry, Pippin and even Sam develop a degree of nobility by the close of the tale, but remain "Hobbitish", while Frodo, almost from the start, seems far different; more "elvish" almost. In many ways, if there were a Hobbit who could wield the Ring, it is Frodo. And in this, we may have a piece of our answer. Perhaps because he may have been, in a sense, "great" enough to wield the Ring, he was ultimately unable to destroy it. This may have been the aspect of Frodo's personality which the Ring used to influence him to claim it.

I have also long argued that Frodo did not fail. He could not throw the Ring into the fireplace at Bag End; how could he have been expected to destroy it. For this reason he was given companions. He was only meant to carry the Ring. He did this very well beyond all reasonable expectations. How Gandalf intended to have Frodo relinquish the Ring so that it could be destroyed is a mystery. Thus Gandalf's "death" in Moria placed the quest in grave jeopardy.