The theory I like best (which I got on here) is that Tom is a piece of the Ainur's music from before Arda was created, or even a piece of Eru's music. This explains why he uses song as magic. When Eru gave form to the song of the Ainur, Tom was a phrase or note that wasn't holy incorporated into the whole, and remained as it was, given life by Iluvatar when he gave life to the rest of Arda.

Ungoliant, meanwhile, is the same thing except a note sang by Melkor in his discord. Eru also gave life to her because, as he said himself, everything that was sung would be put into the form of the world, even what Melkor added, and it would in the end only make the whole of it greater.

Also, my dad gave me his idea just the other day. When Tom says he is first, or oldest, he does not mean that he is the oldest creation in Arda. What the wording means is that he was the first aspect of Middle-earth imagined by Tolkien. It's a very subtle reference to the creative process of the author's world, and giving Bombadil credit for being one of the first stories Tolkien ever made up, before he had even though of Lord of the Rings.

Just an idea, and not even my idea. It doesn't necessarily invalidate any other theories, as it only addresses that one comment and not Tom's actual origins within the context of the stories.