[Guinevere]

Well, as Gandalf said, Bilbo was meant to find the Ring, and Frodo was meant to have it.
So what you really ask is why Eru (or Providence, or whatever...) chose them... Or do you mean why Tolkien, as their creator (and the creator of that world) described the Ringbearer's character such as they are?
I guess it had to be a hobbit, because hobbits do not desire power, as you said. But most ordinary hobbits were quite narrowminded and only concerned with their own affairs, so they wouldn't have been willing to go on a dangerous journey, let alone to sacrifice themselves to save the world. They probably wouldn't even have understood the importance of it all. So it had to be an extraordinary hobbit that had a wider view,a deeper understanding and an interest beyond the boundaries of the Shire. Is that what you call Frodo's "Elvish" qualities? Anyhow, it comes from his beeing taught by Bilbo and Gandalf, as well as his innate charactertraits.
But I think the "Elvishness" grew only in Frodo on the quest. (I'm thinking of Gandalf's remark at Frodo's bed in Rivendell, about the light shining in him, and Sam who sees that light as well)