This is such a great question and was so fun to research! To answer it, we have to work backwards a little bit. First, the only direct answer I could find as to why this game was developed was in an interview with the developer, Richard Robbins, who said he was big fan of the TV show and was inspired to make the game when he had a dream about "Uncle Fester's Playhouse" (building off Pee-wee's Playhouse). Addams Family began as a New Yorker cartoon in 1938, but (as you said) the property gained popularity as a TV show from 1964-8.

But why did the Japanese game studio, Sunsoft, go along with Robbins' idea? Robbins' father, Joe Robbins, was the managing director at Sunsoft, and brought his son Richard in as an executive producer. Joe Robbins had co-owned one of the biggest distributors of coin-operated machines in the U.S. (Empire Distributing based out of Chicago). He sold Empire to Bally Manufacturing in 1972, then went on to work for Atari, where he was able to build connections in Japan and bring Pac-Man over. So Robbins was both high up in the company and had a proven track record of knowing what Japanese game products worked in American markets.

Still, in the interview, Richard Robbins notes that the Japanese executives were skeptical that this "really old weird TV show" would make for a successful game and the Japanese game developers had trouble understanding the dark humor and cultural context of the show. Lynn Spigel has argued that shows like the Addams Family and the Munsters were actually a critique of the suburban values upheld by typical sitcoms of the era. Almost 30 years later, this critique would still prove relevant with the popularity of The Addams Family movie in 1991.

Fester's Quest was successful, despite the game's notorious difficulty, and the younger Robbins went on to develop the Batman NES game, which was also popular.