The first Polynesian-themed bar/restaurant in the U.S. was actually opened in Hollywood, CA in 1934; originally called Don's Beachcomber Cafe, it was renamed to Don The Beachcomber in 1937. At about the same time another Polynesian-themed bar/restaurant, called Trader Vic's, opened in San Francisco, CA. The owners of both restaurants (and, later, chains) enjoyed a friendly rivalry, and both claim to have invented what could be the most famous tiki drink: the Mai Tai.
While we may never know who was the first to mix a Mai Tai, the inspiration for tiki drinks is decidedly Caribbean. References to a type of rum-based cocktail called a swizzle exist in literature as far back as the 1700s. Swizzles usually included some sort of sugar to balance the rum, but were generally an uncomplicated concoction. As the concept migrated north in the 1920s, 30s, an 40s, more complicated combinations of rums, fruit juices, and flavored syrups met involved garnishes... and the tiki drink was born.
Tiki drinks (also know as 'boat drinks' to fans of Jimmy Buffett) gained popularity in the 1940s and 50s, in part due to soldiers returning from war in the tropics and Pacific.
Some of the more famous tiki drinks include the Mai Tai, the Zombie Cocktail (which is the basis for this week's Mike's Mix), the Bahama Mama, and the Hurricane.