Rolex registered the Rolex name for its watches in 1908.
Hans Wilsdorf and Alfred Davis founded their London watch company in 1905 under the name Wilsdorf & Davis. The short, easily pronounced Rolex name became associated with the company three years later, when it was registered for use on its watches.
The company moved from London to Geneva in 1919, helping establish Switzerland as the center of its watchmaking operations. Rolex later became known for milestones including the Oyster case, introduced in 1926, and the self-winding Perpetual movement, introduced in 1931.
A common mix-up is to use 1905 as the answer. That is the founding year of Wilsdorf & Davis, while 1908 specifically marks the registration of the Rolex name.