Athens has sights, of course, but is one of Europe's least attractive capital cities in my view. It is definitely not a place I'd choose for a 2- or 3-week stay (did spend about 10 days there while my mother was hospitalized and recovering from pneumonia). It's not a bad city, and I'm not discouraging a visit. I wouldn't say anything negative at all if you had to base in Athens for work reasons or something like that. It's just that I feel there are so many more interesting cities once you get past the Acropolis and a few museums. Athens is mostly quite modern. Population in 1833 was just 4000.
I haven't been to Nafplion, but a lot of people like it a lot. It's a lot smaller, and I'm not sure what you'd do there after the first 5 or 7 days, but perhaps our experts on Greece will have suggestions. The Peloponnese is very, very interesting; I'm just not sure what's viable as a day-trip from Nafplion. I think there should be some local or semi-local beaches; I think you'd be closer to beaches there than in Athens, but I am not a beach person and don't know for sure. Because of centuries of tourism, English is very widely spoken in Greece. I think you'd find the country affordable, overall, with Athens probably being more expensive than Nafplion.
Istanbul would be a brilliant idea from the sightseeing perspective, and it has both good food and affordability in its favor.
You should be aware, though, that ground transportation from Istanbul or anywhere in Greece to Sofia would be impractical. Your husband would almost certainly have to fly.
In Bulgaria you have two very nice, historic cities that are smaller than Sofia: Plovdiv and Veliko Trnovo. I found them considerably more attractive, but I suppose for a long stay, there should be considerably more to do in Sofia. None of those three cities is near the coast. The small coastal towns of Nesebar and Sozopol are quirky and attractive. They attract the summer beach crowd but are not full of high vacation apartments like you'd find in some other coastal locations. I'd think 2 or 3 weeks in one of those places would be a really long time, though. The food in inexpensive Bulgarian restaurants was rather uninspiring as of 2015; I was lucky that I liked chicken kebabs. Bulgaria is very inexpensive.
Romania is a fascinating country with lovely historic architecture (mainly outside the capital of Bucharest, which has a lot of Ceausescu-era ugly buildings) and some very nice rural areas in the north. However, from the standpoint of choosing just one place to spend 2 or 3 weeks, I can't think of a good reason to favor one of the Romanian cities over a Bulgarian option. (I'm not familiar with the Romanian coast.)
One other possibility would be a spot on the coast of Montenegro. It's a physically beautiful country small enough that some day trips would be possible by bus. The national parks are said to be beautiful, but you'd need a car to access those. Avoid the capital, Podgorica, which is inland and useful only for its airport or as a base for day trips as far as I'm concerned. There's nowhere that would keep you busy for 2 or 3 weeks, but with day trips in the mix, I think it might work; 2 weeks might be smarter than 3. Montenegro's relatively inexpensive.
I'm biased against long stays in most cities (places like London, Paris and Barcelona being exceptions). With that warning, I'll say that the only place in that part of Europe where I'd want to head for 2 weeks or longer is Istanbul. If you're willing to go farther, I'd consider Budapest a good option. It has lots of sights, attractive architecture interesting history, good food and costs lower than cities farther west, but it's nowhere near the beach.
For a quick-and-dirty idea of comparative lodging expenses, you can take a look at booking.com. I've found the reviews on that site pretty reliable; you can't leave a review if you haven't actually stayed at the property.
