From Wikipedia:
Boba is said to have originated in Taichung, Taiwan in 1983, by a Taiwanese teashop owner named Liu Han-Chieh, who experimented with cold milk tea by adding fruit, syrup, candied yams and, finally, tapioca balls. Although the drink was not so popular at first, publicity from a Japanese television show made businessmen take notice and it quickly became well known throughout Asia in the 1990s. In the early 2000s, bubble tea began to become popular in the United States and Canada, spreading to ethnic Chinese, Korean, and Vietnamese-speaking immigrant communities and even outside the overseas Asian community.
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