Character Analysis
The Strawman
Like the Witch, the Scarecrow has a doppelganger in the real world: Hunk, the farmhand who scolds Dorothy about not having any brains.
HUNK: Now lookit, Dorothy, you ain't using your head about Miss Gulch. Think you didn't have any brains at all.
And like the Witch, the Scarecrow is just a flashier version of the same guy. He and Hunk both value brains, and he's so anxious to get them that he's willing to go all the way to the Emerald City despite Dorothy's warning about the Witch. And both characters have an affection for Dorothy that makes their connection very special. (In an early draft of the script, Hunk and Dorothy eventually form a relationship once she gets back to Kansas.) The Scarecrow has the cool floppy limbs and the stunning dance moves, but he's basically still Hunk with a little extra padding.
So yes. Brains. The irony, of course is that the Scarecrow always had brains. He figures things out far more quickly than anyone else in the party, and his quick thinking comes in mighty handy on their adventures. Dorothy needs some apples to eat? Just taunt those living trees to get them! Need a way into the Witch's castle? Beat up the guards and take their uniforms!
This is the guy who needs brains? Seriously Scarecrow, what you really need is a boost of self-confidence.
That self-confidence actually forms the crux of the story for Dorothy's companions. All of them doubt themselves and their capacity, and all of them learn to rely on the qualities they really, truly seem to think they lack. That's why the Wizard can buy them off with a few bits of junk at the end.
The Scarecrow stands at the vanguard of that. "Oh, I'm a failure because I haven't got a brain," he mopes to Dorothy: frustrated that he can't actually scare any crows and blaming it on a lack of brains. Frankly, we're going to blame Oz's crows—we're with Dorothy on the whole "scared to pieces" issue—but the Scarecrow's got this notion in his head, and nothing's going to dissuade him. Nothing, that is, except a long trip with some good folks who really like him, giving him a chance to prove how smart he is even if he doesn't realize it at the time.
In fact, without that wish for brains, the Scarecrow wouldn't have gone on the journey at all. It may have been foolish of him not to realize how smart he really is, but if he were more self-aware, he'd probably still be stuck on the pole in the cornfield. It's his desire to better himself that leads him to confront the dangers of the road, to stick by Dorothy through thick and thin, and most importantly to face his greatest fear: a lighted match. He gets up close and personal with fire a couple of times in the course of his adventure, and learns that he can survive it and even conquer it (with the help of a few buddies and a bucket of water, of course).
That's a lot for a guy made of straw to live up to, but like Dorothy, he does so with flying colors. He might not need brains, but he needs to go on this trip too… and at the end, he's really glad that he did.