Camp at...(listed north to south)
1. Island Pass; explore Marie & Davis Lakes basins
2. Thousand Island Lake on the northwest side; explore Catherine Lake or the upper Garnet basin
3. Ediza Lake; explore Nydiver or Cecile Lake
4. Purple Lake; check out the basin behind it (I'm just guessing here)
5. Silver Pass area; check out Peter Pande Lake, Isaac Walton Pass or somewhere. All of the Silver Divide area is just jaw-dropping colorful and spectacular.
6. Evolution Lake; explore Darwin Bench. But I am one of the rare people who is going to say that Darwin Lakes are overrated, and one of the many who say that Evolution Lake is frightfully overused.
7. Wanda Lake; maybe camp on the southwest side a long walk from the trail; explore Ionian Basin. If you like the idea of staying in Evolution Basin, this option is the more remote one.
8. Muir Pass area; climb the Black Giant
9. Leconte Canyon; hike up to Ladder Lake, requires very good x-country skills. Ladder Lake is really well known for its spectacular nature and is hard to get to. The camping will not be as interesting though (assuming you'll be in a lowland forest in lower Leconte Canyon).
10. Palisade Basin below Ampitheater Lake; hike up to that lake, requires very good x-country skills
11. Lower Palisade Lake, hike over Cirque Pass as mentioned, requires very good x-country skills
12. Upper Basin - you can walk off-trail and just find a little tarn of your own. Gorgeous.
13. Arrow Lake just off the JMT.
14. Upper Rae Lake; explore Sixty Lakes Basin (or Dragon Lake). Rae Lakes is another really gorgeous, but overused area. Consider camping both nights in Sixty Lakes Basin; to get there it's an easy flat-ish hike on a use trail.
15. Bighorn Plateau (in good weather of course)
16. From the JMT south of Bighorn, you can explore Wrights Lake Basin or the lakes at the top of Wallace Creek. But I don't find that section of the JMT (or the camping alongtrail) to be interesting - just the offtrail diversions.
Depends on whether you want to just sort of sit by the lake all day; or whether it's all about the layover day destination. Along the JMT my favorite place to "sit and soak it in" is Sapphire Lake (south of Evolution Lake), but I can't recall if you can camp there, and it is certainly very close to the trail/people. If what you want is to witness the kind of remote and rugged beauty almost no one else sees (and that will be hard to access once off the JMT), I can't think of a better place than a bushwhack up to Ampitheater Lake as a layover day - although I'm not sure exactly where you would camp and how pleasant that camp is. That is as remote/untouched/spectacular as it gets so close to the JMT.
Out of all those, the best place to have a really spectacular camp for 2 nights *and* a spectacular dayhike on your layover day, might be the west/northwest end of Thousand Island Lake. Although that lake is extremely popular, nobody seems to want to walk that far from the trail to the west end of the lake to set up camp. Now you are right under Banner Peak. I would have a very hard time deciding which to do on my layover day: Lake Catherine, or the basin above Garnet Lake. These are jaw-dropping places, both accessible via easy x-country passes, and you won't see anyone unless someone is coming through on the High Route.
Ampitheater Lake, viewed in the distance from Potluck Pass
The idyllic basin behind Garnet Lake and under Banner Peak