Cute as a button
Solution 1:
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.
Solution 2:
Years ago I read in an old volume (early 20th century) the expression "cute as a button quail". It was a children's book originating in the United Kingdom, an anthology of prose, written in a more Victorian dialect. I can't remember the name for the life of me, sorry. I don't know if this usage is related to the "cute as a button" etymologically, but button quails are quite small, and the association with smallness seems to be common. Maybe this will jog someone's memory...
Solution 3:
I found this answer quite intriguing. It comes from http://ndsmcobserver.com/2013/03/cute-as-a-button/.
First, both the words cute and button come to English from Latin by way of French. The etymology of button is as follows: late Latin bottonem became the French bouton and subsequently the English button. The word cute is an abbreviated form of acute, which means small (you may remember something of acute angles in geometry). So, the sources of cute and acute are the same. The precedent for acute is aigu and for aigu, acutus. What does all of this mean? Well, it turns out that in addition to the meaning of small, aigu is often used in medical terminology to mean a condition that appears abruptly and needs urgent care, which is also true of the English acute. This seems irrelevant until you consider that bouton can refer to a pimple or spot. Finally, the term “cute as a button” is known to have arisen in the 1800s, a time when diseases like chicken pox, measles, mumps and the dreaded small pox threatened lives daily. Now, we bring it all together. In a time when chicken pox and other diseases ran rampant, a word recently derived from another word used for sudden symptoms and the need for urgent care and a word derived from another word used for dermatological spots find themselves in the same unexplainable colloquialism. To state what is by now obvious, “seeming as in need of medical care as someone with spots from chicken pox, measles, etc.” was the original meaning and proper usage of “cute as a button.”
Solution 4:
I found this link on a site called funtrivia.com, And I can certainly see where it would make sense, especially as the woman explains it as a shortening of an original phrase, much like ‘happy as a clam’ for ‘happy as a clam at high tide’.
“I found another explanation here: The origin of the phrase "cute as a button"...it actually has nothing to do with a button as in a button on a shirt. the button quail is a very small gray super, super fluffy squishy looking (or 'cute' if you will) bird. people used to say "cute as a button", meaning "cute as a button quail" b/c the bird was considered so adorable.
link https://robingriggswood.wordpress.com/2013/12/02/robin-griggs-wood-google-cute-as-a-button-ever-wonder-where-that-phrase-comes/“