Upvoted for the private island idea.
If you're anywhere NYC, you could perhaps check out the Adirondacks. The camping season isn't open up there yet, but if you're looking for peaceful tranquility, peace, time to reflect, and the inspiration of a glorious nature scene, then I highly recommend the private island campsites on Lower Saranac Lake.
They're muni-run camp sites, so the costs are very low -- I think we paid $22 a day. The one catch is that they fill up quickly, and they're hard to get.
There are island camps with single occupancy, which is to say that there is only one camp site on it. And there are larger islands that are multiple occupancy, though we did the math, and even on the most crowded of them, the campsites are spaced quite far apart, and you will have at least an acre to yourself (approximately). We chose the single occupancy, and loved every second of it.
It's made much better if you own your own canoe or fishing kayak, otherwise boat rentals are fairly high. You CAN take a taxi out to the island and have them pick you up, but I wouldn't do this at the very start of the season or the very end. There isn't enough traffic there until peak season starts to ensure that somebody would be around if you got hurt, or were mauled by a bear or something.
This link has plenty of detail: http://www.dec.ny.gov/outdoor/24496.html