It seems that the word Bollywood derives from Tollywood.
Perhaps it’s worth mentioning here that ‘Tollywood’, as a name for the Calcutta film industry was in circulation long before anyone had ever heard of Bollywood. The term is a portmanteau, Hollywood and Tollygunge (the area in Calcutta where many film studios were located) packed up into one word. JS magazine used it liberally in the late 60s and early 70s, so it isn’t unlikely that someone thought up ‘Bollywood’ as a variation on the joke about the same time.
Madhava Prasad digs even further, and traces the origin of Tollywood to an American engineer who introduced talkies to India in 1932:
In 1932, Wilford E. Deming, an American engineer who claims that ‘under my supervision was produced India’s first sound and talking picture’, writing in American Cinematographer (12.11, March 1932), mentions a telegram he received as he was leaving India after his assignment: Tollywood sends best wishes happy new year to Lubill film doing wonderfully records broken. In explanation, he adds, ’In passing it might be explained that our Calcutta studio was located in the suburb of Tollygunge… Tolly being a proper name and Gunge meaning locality. After studying the advantages of Hollygunge we decided on Tollywood. There being two studios at present in that locality, and several more projected, the name seems appropriate.