While I can't find any scholarly answers, most answers I'm finding say that 'button' refers to something pretty or attractive in a dainty way. After all, you're using the word 'cute' so you wouldn't be using it to describe a large, muscular man. This phrase would be best suited for a small child or flower.

> CUTE AS A BUTTON - "cute, charming, attractive, almost always with the
> connotation of being small, 1868 (from the original 1731 English
> meaning of 'acute' or clever). Cute as a bug's ear, 1930; cute as a
> bug in a rug, 1942; cute as a button, 1946. Cute and keen were two of
> the most overused slang words of the late 1920s and 1930s." From
> "Listening to America" by Stuart Berg Flexner (Simon and Schuster, New
> York, 1992.)
> 
> Flexner may have an idea about the word "cute," but he provides no
> guidance on the question of how a button can be cute. The key to the
> issue is that it is not the button on a shirt that is meant here, but
> a flower bud seen in the popular name of small flowers, such as
> bachelor's button (q.v. "button" (n) in the OED, meanings 2 and 3).
> 
> The British version is "bright as a button". This makes sense if you
> think of a polished brass button. The phrase is really only ever used
> of small people - you'd say that a child, or maybe a small dog, was as
> bright as a button, but you'd never say it of a six-foot man. So the
> image is of a small sparky thing.
> 
> http://www.phrases.org.uk/bulletin_board/41/messages/652.html

You can read more [here][1].


  [1]: http://www.answerbag.com/q_view/184286#ixzz1xyWiXn5n