The answer to the question: What lake island is farthest from shore?
First, some clarification. I'll accept any lake including saline or endorheic lakes. By farthest from land I mean if you take the shortest distance between any point on that island to any point on the border of the lake, what is the longest resulting distance you might have? This would count against a large lake island whose far side was far from the lakeshore, but might have a near side very close by (e.g., Manitoulin for example). Secondly I do mean from the lake shore, not closest other land (which might well be another island of a chain).
Right off the batch I started looking for large, wide lakes, as opposed to long narrow ones. e.g., Lake Tanganyika is a very large lake, but you can't get that far from shore because it is a very long, relatively narrow lake.
The Caspian seemed a logical place to start, but its islands seem to be entirely coastal. Kulaly Island seems to be the farthest from shore I could find.
There are two other logical locations to search: The Great Lakes of North America... and Lake Victoria in Africa.
First, the Great Lakes. Glancing at a map there are a few large islands that jump out immediately, Isle Royale in Lake Superior... and the Beaver Island Archipelago in Lake Michigan. Isle Royale is a little to close to shore, the others looks a little more interesting. There are a number of islands surrounding Beaver Island, and trying to determine which is farthest from land is a bit tricky. Actually the nearby Fox Islands might be better.
However, my candidate for lake island farthest from shore in the great lakes is Caribou Island in Lake Superior. There are actually several islands with that name including at least one other in Lake Superior, in the Thunder Bay area. But the one I mean is located at 47° 21′ 33″ N, 85° 48′ 36″ W.
Caribou Island appears to be about 30 miles from the closest shoreline of Lake Superior, which is to the East-Northeast in Superior Provincial Park in Ontario. There is a closer shoreline in the form of Michipicoten Island, but as I noted we're not counting other islands.
Now as for Lake Victoria... it is a very large lake (as well as generally acknowledged as the source of the White Nile). There are many islands in the lake... At first glance closest to the center appears to be Nabuyongo Island. At second glance it seems a peninsula from the southeastern end of the lake extends towards it reducing its distance... but looking closely you see the end of that peninsula is actually an island itself (named Ukerewe), separated by a narrow channel. A quick eyeball test puts it at between 50 and 60 miles from the closest point on that peninsula.... and about the same distance to the western shore of the lake.
Anyway this is my candidate for lake island farthest from shore in the world. I havent' gone and calculated actual distances via GPS coords or anything, the difference seems to be large enough to justify it as nothing else I've found comes close. Anyone have another candidate worthy of consideration?
FWIW Nabuyogo Island was the site of a naval battle that took place on Lake Victoria during World War II.