Destin is named for the fishing captain who settled it. Leonard Destin (1813–1884) came from New London, Connecticut, to the shore of East Pass, where Choctawhatchee Bay meets the Gulf of Mexico. Sources give different dates for his arrival, from the 1830s to between 1845 and 1850. He and his wife, Martha McCullum Destin, built a fishing business that employed young men from the area, and many of those workers became the community's first settlers; according to the Destin History and Fishing Museum, 16 founding families still live in the area. Destin built a New England colonial-style home on the site later known as the Moreno Point military reservation, and he and his descendants fished these waters for generations.
More about Destin
The pass that defines Destin today was born in a storm. In 1929, according to the museum's director, a storm pushed Choctawhatchee Bay about eight feet above normal. To relieve the flooding, four local men, O. T. Melvin, Dewey Destin Sr., Arn Strickland and Dolf Weekly, dug a trench across the sand with shovels and boat oars, and within a couple of days the rushing water had widened it to about 500 feet. The new pass closed off the old one to the east, turned the mostly fresh bay salty and gave Destin's fishing boats a direct route to the Gulf. Some local accounts date the event to 1926.
For much of its history Destin was a small, isolated fishing village. Its first condominiums went up in the 1970s, the city incorporated in 1984, and it has grown rapidly since, while still calling itself "the World's Luckiest Fishing Village."
Sources
Wikipedia: Destin, Florida
Emerald Coast Magazine: Take a Step into the Past (Destin History and Fishing Museum)
Get the Coast: Four men, shovels and a storm: the accidental birth of Destin's East Pass (citing Vivienne Williams, Destin History and Fishing Museum)