Introducing Sir Turtle, the man of the island, he adorns nearly everything here, so I thought I'd introduce him to you. I've been doing some research and it seems this was designed by Suzy Soto (the above is my sketch of her design) in 1963 and sold to the Department of Tourism in 1970 for the princely sum of $1 but despite this, the above design is only the 'unofficial' national logo.
If you arrive in George Town on a cruise liner you'll meet Sir Turtle at the Port Authority, by plane and he graces Cayman Airways (with the addition of a flying scarf), at every turn there is he is in all his pirate-y splendour.
It's a pretty interesting and complex design for a national logo, unofficial or otherwise, an ear-ringed pirate turtle with a wooden leg, imagine a design team convincing a tourist board of that today!
The Cayman Islands were originally named Las Tortugas by Christopher Columbus after the numerous sea turtles, and Tortuga is the name of the island rum (and the tourists' favourite Tortuga Rum Cake).
So there you have it, a little island knowledge.