Mental Floss explains how each Major League Baseball team got there name. Here's the Red Sox story:
The team that became known as the Red Sox began play "“ wearing dark blue socks, no less "“ as a charter member of the American League in 1901. With no official nickname, the team was referred to by a variety of monikers, including Bostons and Americans, as in American League. In 1907, Americans owner John Taylor announced that his team was adopting red as its new color after Boston's National League outfit switched to all-white uniforms. Taylor's team became known as the Red Sox, a name popularized by the Cincinnati Red Stockings from 1867-1870 and used by Boston's National League franchise from 1871-1876.
The Dodgers story is pretty funny:
The Dodgers trace their roots to Brooklyn, where the team was known as the Bridegrooms, Superbas, and, beginning in 1911, the Trolley Dodgers. The Dodgers nickname referenced the pedestrians who dodged the trolleys that carried passengers through the streets of Brooklyn. While the team was known as the Robins from 1914 to 1931, in honor of legendary manager Wilbert Robinson, the nickname switched back to Dodgers when Robinson retired. When Walter O'Malley moved the franchise to Los Angeles after the 1957 season, he elected to keep the name.