Saturday lunch near Mystic Seaport
I'd highly recommend Pita Spot which is less than a mile from the Seaport. They serve great Lebanese food at very reasonable prices, have incredibly friendly service, and pleasant decor. It's BYOB and there's a package store right across the street.
I've had many good meals (and some not so good) in Mystic, but most of the restaurants have been rather casual and family friendly. There is a nice restaurant that overlooks the drawbridge in Downtown where I've had wonderful meals and enjoyed the nice atmosphere. The problem is, Mystic is heavily geared towards families, and the restaurant choices reflect this. Additionally, there seems to be huge restaurant turn-over. There one day, gone the next. At least, that's my opinion.
Stonington, a short drive from Mystic, has some grown-up restaurants. You might try Noah's on Water St.
If you are looking for a romantic meal, and you can drive a ways, try the Copper Beach Inn in Ivoryton, or the Bee and Thistle in Old Lyme. But reservations can be difficult to get, so call early.
I wish I could have been more helpful-there are so many restaurants in the Mystic area, but I have kids (and no s.o.) so hopefully someone else will offer tips, too.
Before dinner, if you have time, visit Jonathan Edwards Winery in N. Stonington- the wines are good, and you can sit and chat.