her name. She begins to scandalize European women immediately by arranging to take a trip with Winterbourne to the famous Castle of Chillon, immortalized by Lord Byron in his poem "The Prisoner of Chillon." She doesn't know this man at all. It is shocking for her to take a day-trip with him unchaperoned. Winterbourne himself doesn't know what to make of her. As seen from his viewpoint she is fresh and honest, very independent and self-reliant, very trusting, unlike European girls of her class and age.

...and then he saw that this glance was perfectly direct and unshrinking. It was not, however, what would have been called an immodest glance, for the young girl's eyes were singularly honest and fresh. They were wonderfully pretty eyes; and, indeed, Winterbourne had not seen for a long time anything prettier than his fair countrywoman's various features--her complexion, her nose, her ears, her teeth. He had a great relish for feminine beauty; he was addicted to observing and analyzing it; and as regards this young lady's face he made several observations. It was not at all insipid, but it was not exactly expressive; and though it was eminently delicate, Winterbourne mentally accused it--very forgivingly--of a want of finish.

Winterbourne wishes to introduce Daisy to his Mrs. Costello, his aunt, in order to give a certain air of formality and respectability to this impromptu relationship. But she refuses to meet any member of the family. She judges Daisy with the severity which most European women will duplicate in other countries.

"They are very common," Mrs. Costello declared. "They are the sort of Americans that one does one's duty by not--not accepting."

The nickname "Daisy" also suggests fragility, since a daisy is a short-lived flower, just as it is undeniably pretty and common. It is also commonly plucked of its leaves by women's fingers. The story has to end with the girl's death, which comes about because of her naivete and her characteristic American insistence on life, liberty and the pursuit of happiness. As described in Critical Survey of Literature for Students in the eNotes Study Guide:

One night, despite the danger from malarial fever, Giovanelli takes Daisy to the Colosseum. Winterbourne, encountering them in the ancient arena, reproaches the Italian for his thoughtlessness. Giovanelli says that Daisy insisted upon viewing the ruins by moonlight. Within a few days, Daisy is dangerously ill. During her illness, she sends word to Winterbourne that she was never engaged to Giovanelli. A week later, she is dead. 

Daisy Miller is a fairly characteristic Henry James story contrasting American and European cultures. Daisy is an innocent girl--but the European women are naturally suspicious because there is so much more immorality in their own culture. They are actually judging her by themselves. They all reject her. One woman actually insults her outrageously at a social function. It is understandable that Daisy would be thrown more and more into the company of men, especially Winterbourne and the worldly wise courier Giovanelli. 

Henry James's novella Daisy Miller has often been compared/contrasted to F. Scott Fitzgerald's realist novel The Great Gatsby (see sidebar). It is perhaps in this comparison that one discovers several clues to the possible meanings of protagonist Daisy's name. Another interesting comparison is this title character to Maisie in James's What Maisie Knew.

Consider the billboard in The Great Gatsby and the "all-seeing eye of God" represented by the spectacles. If Daisy's name is derived from "Day's Eye" then this character sees and understands more than her antagonist Winterbourne perceives - just as Maisie sees far more than her parents realize. Additionally, her name symbolizes the link to the natural world. The polarity between Daisy's name and Winterbourne's also parallels the differences between American and European culture and mores.

Not much has been written about Daisy's surname; however, consider the drone of the mill and the Daisy's droning on about American Society (see link below).

As you explore this question further, consider any emotional components to the name or any feelings her name elicits. And what is the life span of a daisy? Finally, be sure to examine carefully how the tropes outlined in this answer ultimately relate to the enduring questions Henry James raises in this novella as well as in his other seminal works.

See also:

http://blogs.baruch.cuny.edu/americannovel/?p=84

http://www.sheknows.com/baby-names/name/daisy