There is a very good book called And a Bottle of Rum by Wayne Curtis that talks about this very question! It's a history of not only rum, but how the United States was shaped by it. I'll summarize the story here:
While there were experiments in fermenting and distilling sugar around the world, rum was initially a way of making use of waste products of sugar refining. It started out as a pretty awful drink, once called a "hot, hellish, and terrible liquor" and nicknamed "kill devil." Despite the popular association of rum with the Royal Navy, it was initially a sort of substitute standard for beer, wine, and brandy when resupplying in the Caribbean. It was also shipped up north and became a very cheap imported liquor, equally useful for getting drunk as trading with the Indians and making them more susceptible to suggestion during negotiations.
Rum started to improve as it began being manufactured in the North American colonies, who took advantage of very cheap molasses from the French Caribbean (who abused slave labor to an unfathomable and horrendous degree to create massive sugar plantations). There was concern about alcoholism and abuse of hard liquor as opposed to such saintly drinks as beer and wine, but rum took on an identity as a colonial drink. By 1770, the Thirteen Colonies were producing about 5 million gallons of rum per year.
Parliament fought the heavy buying of French molasses with taxes that were widely ignored, worked around, or avoided completely through smuggling. The Sugar Act of 1763 is covered in school as one of the examples of taxes that angered the colonists on the road to independence, but it's not as often covered that the act was directly related to the huge rum industry in the colonies. It was a blatant effort by the British government to give an advantage to their own West Indies sugar plantations, a classic example of capitalist influence on politics at the cost of the colonial producers. While the act was amended in 1766 to reduce the grumbling, the arguments over the sugar tax was one of the earliest examples of American resistance to their lack of representation.
After the war for independence, however, rum suffered a steep decline. Sugar plantations in the West Indies became less profitable due to overproduction expending the soil and Britain emancipating its slaves, while the British colonies obviously had some more trouble trading with the newly independent United States of America. At the same time, expanding agriculture as Americans moved west created huge amounts of cheap grain (namely maize) that could be distilled into whiskey. Rum lost its status as the cheap go-to liquor in favor of whiskey in the 19th century.
What gave rum a comeback, ironically, was Prohibition. The Caribbean was very open to American tourists with lots of money looking for a nearby place to legally drink, and Bacardi in Cuba had just begun producing its famous light rum that was an easy mixer. Americans sailed and flew by the millions to the islands, especially Cuba. There are accounts of boats approaching ships as soon as they came within sight of shore to try and sell bottles of cheap rum to passengers and crew. Rum drinks like the Daiquiri became popular and fashionable until they were known worldwide by the end of World War II. The closeness of these islands also meant tons of smuggling across the Gulf of Mexico and the Atlantic.
However, this came at the price of variety. Light rum like Bacardi made was the new trend, establishing rum as a generically sweet liquor with little flavor to mix into whatever. The tiki craze established by Donn Beach and Vic Bergeron tried to bring different varieties and flavors of rum to the front of complex cocktails, but these were quickly distorted into generic fruity drinks that completely covered the taste of any alcohol with them (I seriously implore you to investigate classic 1930s tiki recipes because they are nothing like you would find at Bahama Breeze). While rum had made a resurgence in America, it had done so in its worst possible form.
Thankfully, the general explosion in craft everything in the last 20 years has made life a lot easier. There's now far more rum variety on the market and rum has become almost as well regarded as whiskey as a sipping spirit, with bottles costing over $100 regularly appearing on market shelves in areas with good rum distribution (Florida, NYC, the West Coast, and Chicago are the big areas for rum from what I've been told by my expert friends; I live in a small town just north of Orlando and I can buy over 200 rums from nearby liquor stores). There's also been an increase in interest in vintage drinks, including colonial punches and the tiki drinks of yesteryear, which has greatly helped rum return to form. Hopefully it stays that way.