Wikipedia’s history of Bollywood states: The name “Bollywood” is derived from Bombay (the former name for Mumbai) and Hollywood, the center of the American film industry.
The naming scheme for “Bollywood” was inspired by “Tollywood”, the name that was used to refer to the cinema of West Bengal. Dating back to 1932, “Tollywood” was the earliest Hollywood-inspired name, referring to the Bengali film industry based in Tollygunge, whose name is reminiscent of “Hollywood” and was the center of the cinema of India at the time. It was this “chance juxtaposition of two pairs of rhyming syllables,” Holly and Tolly, that led to the name “Tollywood” being coined. The name “Tollywood” went on to be used as a nickname for the Bengali film industry by the popular Kolkata-based Junior Statesman youth magazine, establishing a precedent for other film industries to use similar-sounding names, eventually leading to the term “Bollywood” being coined.
The term “Bollywood” itself has origins in the 1970s, when India overtook America as the world’s largest film producer. Credit for the term has been claimed by several different people, including the lyricist, filmmaker and scholar Amit Khanna, and the journalist Bevinda Collaco.