Wordorigins.org

It is often thought that the name of this famous cocktail comes from the brand name Martini & Rossi, makers of vermouth. While the Martini & Rossi brand influenced the naming of the drink, it is not the ultimate origin.

Instead, martini is a toponym, named after its place of invention, Martinez, California (a small city northeast of San Francisco). It was originally known as the Martinez cocktail, a 1:2 mix of gin and sweet vermouth with maraschino cherry juice and bitters. From O.H. Byron’s Modern Bartender’s Guide of 1884:

Martinez cocktail...Same as Manhattan, only you substitute gin for whisky.

The name had shifted to martini by 1887. From the Brooklyn Daily Eagle of 30 June of that year:

Sample the bewildering depths of the “Martini cocktail.”

By 1903, the dry martini that we are familiar with today had come into existence.

The first US trademark for Martini & Rossi was filed in 1882. The brand name undoubtedly influenced the shift from Martinez to the modern martini.

(Source: Oxford English Dictionary, 2nd Edition)