Maybe you haven't been to/heard of Vevay (it's lovely this time of year!) but you certainly have some pretty strong associations with Rome. They say it wasn't built in a day, and the architecture really shows the time commitment. One thing's for sure: these places are classic and steeped in mythic import.
The castle at Chillon near Vevay, where Daisy and Winterbourne go on the day trip that sparks their romance, is the setting of a dark poem about a prisoner by Lord Byron, one of the most popular poets of the Romantic period.The French philosopher and novelist Jean-Jacques Rousseau also wrote a hugely popular novel in the late 18th century that was set in Vevay. Called Julie, it's about first love between two young innocents.
Rome is where some of the most dramatic stories in all of literature go down, ranging from Shakespeare's Antony and Cleopatra to Hawthorne's The Marble Faun. In a sense, Daisy Miller is set in the world of literature as much, or even more than, it is set in the real world.
James picks places that remind us of stories so that we already sort of know how we're supposed to feel when we get there. It's like how when characters in a movie go to the top of the Eiffel Tower or the Empire State Building, you know they're going to kiss and drop some huge romance bomb—usually an engagement. Oh, and there's also the "when in Rome" aspect of Rome. You know, you're supposed to do as the Romans do. Or, in this case, as the Americans in Rome do.
Rome and Vevay are specific locations in Europe, and a lot of this novel is about the contrast between America and Europe in general. America is the New World and Europe is the Old World. Americans go to Italy to learn about art, to Paris to buy clothes, to Switzerland to take in the healthy air, and to England to eat meat pies and bad-mouth the monarchy.
New money people like the Millers don't have a good, classical education or a strong foothold in high society. Europe is the perfect place to cultivate both of those things. So Daisy is supposed to be there to become best buds with people like Mrs. Costello while roaming through cathedrals and art galleries. But we all know she's more interested in sampling the living Italians than their musty oil paintings.