I like your hammock idea. I always put an old fashioned sleeping bag (rectangle lined with flannel) on the hammock so it is comfortable and in the fall you can creep inside it to stay a bit warmer.
I also like the notion of a shed so there is the seasonal appreciation of closing things up for the winter (place for lawn chairs and mowers, etc.). There is something emotionally significant particularly for kids of opening and closing things for a season.
I would also go to a thrift store and buy a bunch of books (in addition to the board games and cards mentioned above). I would get books of interest across an age range for rainy days and also get some field guides to birds, animals native to the area, insects and plants.
Last, but not least, a good first aid kit that is a bit more expansive than you might have at home (I am assuming you are further away from medical care than when at home). This will need to be restocked every season so it is ready to go when you need it.