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
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