Since buttons aren't particularly cute (IMO), where did this common phrase come from? I know it's old; I've seen it in 19th century literature.
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.
You can read more here.
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...
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.”
I can think of three explanations, not necessarily mutually exclusive:
Cute originally meant smart or clever. The button, as a fastening invention, is quite ingenious. As cute came to mean "attractive" or "pretty", the phrase moved in meaning as well.
Buttons were small delicate and elegantly decorated, comparable to (say) a cute doll.
It's deliberately a nonsense phrase, as with some postulated origins for "dead as a doornail". I imagine it being invented by the same sort of people who came up with "madder than a wet hen", "two days older than dirt" and the like, but perhaps that's just idle fantasy.
Although I agree with the author of the Phrase Finder remarks (cited in skarz's answer above) that two other phrases—"cute as a bug's ear" and "bright as a button"—are highly relevant to an inquiry into the origin and sense of "cute as a button," I disagree with several of his or her conclusions.
When did the three phrases originate?
First, the dates given for all three phrases are off by a significant amount: Phrase Finder dates "cute as a bug's ear" to 1930 and "cute as a button" to 1946; it doesn't give a date for "bright as a button," but describes that phrase as "the British version" of "cute as a button," which many people (I imagine) would interpret as indicating that it is roughly the same age as "cute as a button."
A Google Books search, however, finds instances of "bright as a button" going back to 1805. Here are the earliest three, in chronological order. From "Yankee Phrases," a poem consisting largely of similes drawn from the northeastern United States, in The Spirit of the Public Journals; or, Beauties of the American Newspapers, for 1805 (1806):
She smil'd like a basket of chips—
As tall as a may pole her size—
As sweet as molasses her lips—
As bright as a button her eyes.
From William Stone, "The Mysterious Bridal," in Tales and Sketches: Such as They Are, volume 2 (1834):
"Now that's what I call a little too slick—its true poetry—and that's what can't often be said, I calculate," exclaimed Mike, in honest admiration. "I can remember jest how Birgwine looked there at Saratoga, when he marched out to give up his toasting iron to Gates. But there was Schuyler, and Brooks, and Lincoln, who, I'm plaguy fear'd, is now coming arter us. They was all ra-al fighting characters. And then there was, Arnold, too—his eye was as bright as a button—he was a ra-al fire-eater—but then he was a desput traitor"—
And from Nathaniel Hawthorne, "The Story Teller" (later republished as "Mr. Higginbotham's Catatrophe") in The New-England Magazine (December 1834):
Meantime, Dominicus Pike, being an extremely polite young man, and also suspecting that a female tongue would tell the story as glibly as a lawyer's, had handed the lady out of the coach. She was a fine, smart girl, now wide awake and bright as a button, and had such a sweet pretty mouth, that Dominicus would almost as lieves have heard a love-tale from it as a tale of murder.
Notably all three of these instances are from the United States, as are additional matches from 1835, 1841, 1857, 1863, 1869, and later. The earliest British English instance of "bright as a button" is from Rudyard Kipling, "Private Learoyd's Story" (1888), in Plain Tales from the Hills, 1886-1887: Soldiers Three and Other Stories (1899).
So whether "bright as a button" was ever the British version of "cute as a button" it appears to have begun as an American saying, no later than the very early 1800s.
The phrase "cute as a bug's ear" has a first occurrence date in the Google Books database of 1913. From "Long Island Sound Championship," in Forest and Stream (January 18, 1913):
Much of the credit for the expert handling of the event is due Jim Aiker, whose score fell far below his average because of his interest in everybody else. Carl von Lengerke, representing “If it's black, it's us,” and Brother Keller, of the “yellow boys,” helped considerably. Then, to add to the joy of the occasion, a damsel from Mt. Vernon, Miss Thorpe, cute as a bug's ear, shot fast enough to trim her father, though some uncharitable party said "pop missed the last four on purpose." Be that as it may, the little lady lent a willing hand and a pretty face.
The phrase "cute as a button" debuts in Google Books in 1931. From a letter of Christmas shopping hints dated December 1, 1931, in Indiana Telephone News, volume 21 (1931) [combined snippets]:
I always can tell what the kids will like by the way George behaves. That man will never grow up! This year he had a fit over a little Coca-Cola truck (only 49c, imagine!) with ten miniature coca-cola bottles just as cute as a button, and a foolish little box of figures called Krazy Ikes for a dollar.
To recap, the earliest Google Books matches for the phrases are "bright as a button" 1805, "cute as a bug's ear" 1913, and "cute as a button" 1931. "Bright as a button" is more than a century older than either of the other two, and "cute as a bug's ear" is about 18 years older than "cute as a button."
What do the the phrases mean?
Christine Ammer, The Facts on File Dictionary of Clichés, second edition (2006) has this entry for "cure as a button":
cute as a button Daintily attractive. The word "cute" dates from the seventeenth century. It was originally an abbreviation of acute and had the same meaning: clever, shrewd, ingenious. In America, however, it came to be applied to attractive persons or things, those with an appearance of dainty cute objects. For some reason this simile took hold in the early twentieth century. The synonym cute as a bug's ear similarly alludes to something very small—and in fact nonexistent (bugs don't have ears).
So (according to Ammer) "cute as a button" means "daintily attractive." Is that what "bright as a button" means in British English? I see no reason to think so. John Ayto, The Oxford Dictionary of English Idioms, third edition (2009) has this entry for "bright as a button":
bright as a button intelligently alert and lively. informal [Note:] There is a play here on bright in its Old English sense of 'shiny' (like a polished metal button) and bright in its transferred sense of 'quick-witted', found since the mid 18th century.
The same double meaning is discussed in a memorable exchange between the Scarecrow and a lost (and not very well-informed) little boy named Button-Bright in L. Frank Baum, The Road to Oz (1904):
While they waited, the Scarecrow, who was near the little boy, asked:
"Why are you called Button-Bright?"
"Don't know," was the answer.
Oh yes, you do, dear," said Dorothy. "Tell the Scarecrow how you got your name."
"Papa always said I was bright as a button, so mamma always called me Button-Bright," announced the boy.
"Where is your mamma?" asked the Scarecrow.
"Don't know," said Button-Bright.
"Where is your home?" asked the Scarecrow.
"Don't know," said Button-Bright.
"Don't you want to find your mamma again?" asked the Scarecrow.
"Don't know," said Button-Bright, calmly.
The Scarecrow looked thoughtful.
"Your papa may have been right," he observed; "but there are many kinds of buttons, you see. There are silver and gold buttons, which are highly polished and glitter brightly. There are pearl and rubber buttons, and other kinds, with surfaces more or less bright. But there is still another sort of button which is covered with dull cloth, and that must be the sort your papa meant when he said you were bright as a button. Don't you think so?"
"Don't know," said Button-Bright.
Where did the phrase 'cute as a button' come from?
"Cute as a button" originated in the United States, as did "bright as a button" and "cute as a bug's ear." Since both of those other phrases were well established in U.S. English at the time that "cute as a button" emerged, I think it isn't at all far-fetched to suggest that "cute as a button" may be the product of a merging of the daintily attractive "cute" from the "bug's ear" simile and the small but visible "button" from the "bright as" simile.
The Phrase Finder commentary that serves as the basis of skarz's answer above was written in 2005—before Google Books Ngram searches were available, I believe—and in any event seemingly without reference to any other searchable database of texts that might have turned up more-accurate details about how old the three phrases are and where "bright as a button" originated. It's still an interesting answer, but I think its conclusions are flawed and out-of-date.