Peach Melba is named after a singer, and the singer was named after a city. Follow the dessert back far enough and you reach Melbourne.
Nellie Melba was the stage name of the Australian soprano Helen Mitchell (1861-1931). She took "Melba" from her hometown. Melbourne itself was named in 1837 for the British prime minister Lord Melbourne, whose title came from an English place name meaning "mill stream."
The standard account of the dessert begins at the Savoy Hotel in London in 1892 or 1893. Melba was singing in Wagner's Lohengrin at Covent Garden, an opera with a swan in it. Escoffier, the Savoy's chef, served her peaches on vanilla ice cream in a silver dish set on an ice sculpture of a swan. He called it pêche au cygne, "peach with a swan."
The raspberry came later. Wikipedia's account says Escoffier added sweetened raspberry purée after he and César Ritz moved on to open a new London hotel, and only then named the dish for Melba.
Escoffier's own recollections tell it differently. In a 1908 newspaper interview and a 1925 trade article, he linked the idea to the Ritz in Paris. He placed its public debut at the opening of the Carlton Hotel in London. His biographer Kenneth James, quoted by Barry Popik, notes that Escoffier said the dish did not grow directly out of the swan version. Melba herself remembered first being served it at the Savoy. The route from swan to raspberry is uncertain, but the Melba connection is not.
The printed record starts in 1902. An American book that year asks, "Did you ever hear of peche Melba?" and the Philadelphia Inquirer printed a recipe for "Peaches Melba." Escoffier included the dish in his Guide Culinaire of 1903. Etymonline dates peach Melba "by 1905." American soda fountains were selling a "Peach Melba" sundae by 1906.
The singer lent her name to a second food as well: Melba toast, recorded by 1913.