What was the origin of the Hobbits?

Among the many "origin" questions about Tolkien's work, I am surprised this one hasn't come up (at least, not more often)

What was the origin of the hobbits? Not the origin of why Tolkien started writing about them, but their origin in the story.

Hobbits are cosmologically humans, or Hildor--- they have the Gift of Men, leave Arda, and are moral. (And can become sick---Bilbo's cold is actually an important cosmological key!). This is confirmed in the writing, Hobbits are a type of human. But they differ in size, appearance and temperament. When did Hobbits split from the main branch of men, and why?

There are two options:

  1. Hobbits were part of Eru's plan all along, and they woke at the same time as the other Hildor, in FA1, in the form of Hobbits already.
  2. Hobbits "evolved" from humans of normal stature and temperament, sometime in the First or Second Age.

There is no evidence either way for these options, and there is 5000 years of history, between FA 1 and TA 1000, when Hobbits could have arose.
There is evidence in Tolkien of both single individuals (Ents becoming treeish) and races (Petty Dwarves) changing form and size, so it could be that is what happened to Hobbits.

But it is interesting, before I started thinking about this, I realized I had always assumed that Hobbits changed form some time in the intervening period. Like, I guess my vague thought was (and still is), that Hobbits were men who did something noble, or just needed to escape, and so a Vala (like Yavanna) asked Eru to grant them small size. That is how I imagined it, but I can't justify that.