Lac la Belle Lodge and Grill in Lac La Belle has excellent food. There was also a place in Lake Linden I think the Dugout II or something like that. They had a great deal on steaks a few times we were there. Right on trail into town. gas right across the street.
I wasn't going to go that far but I do recommend the Mosquito as well. Parkview Lodge has a nice eatery as well but my favorite if you go all the way to the tippy-top of the island would be the Mariner North in copper Harbor.