Flinders Island is a top place to stay. Middle of the Island has the town (capital) of Whitemark and the place to stay is the Interstate Hotel....good rates and top meals.
South end is Lady Barron (strip as well) and the place to stay there is the Fernaux Tavern. The Northern end has been covered in thread above. (Killercrankie).
Overall, the Island is a top destination and is worth a 10/10 for everything! Have fun and don't be afraid to stay awhile.
Next Island down is Cape Barron - also one very special place but alas, not much in the way of accommodation. Certainly worth a visit though. Next Island south is Clarke Island and that is pretty much off-limits with no accommodation or facilities at all.
Some of Australia's best kept secrets are in this neck of the woods so psssst; don't tell everyone eh?