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
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.
You can read more here.