It's an interesting place in history, but the origin of the name is undetermined. It has nothing to do with jam or raspberry.
In Danish the Island is known as "Sild" meaning herring and in German it's known as "Sylt" which again used to be an old Danish word for low land salty marsh, which it very much is (or at least used to be). The herring is part of the coat of arms of the island, so it's likely that Sild and Sylt are the same word.
The viking invasion of England set out from this island around year 450.
With the Nordic population moving to the British isles the island was left uninhabited for hundreds of years, until the Frisians moved there around year 800, naming it "Söl". Again, this is prior to written definitions, so it's likely that Söl is the same word as Sild and Sylt...
In 1362 the entire area was flooded in The Great Drowning of Men, changing the landscape to it's current island form.
Denmark lost the island to Germany in 1864 and despite the island having its own Frisian dialect it took the official German name "Sylt".