Busan! My boyfriend and I did a day trip, but I would definitely recommend staying overnight since the night views are the best.
There are other options, too. Like as ethelman2 recommended, Gyeongju. If you're more into modern history, Gwangju (in the South, nearby Jeolla province) is also a good choice. It's where the Gwangju uprising took place. There are a lot of great history museums and sights to be seen.
Gangwondo is also nice if you want to stay within driving distance of Seoul, but want to see the ocean and beaches. You can also see the mountain (Seoraksan) and do either a cable car or hike to the summit.
If you want something more off the beaten path, I recommend South Jeolla (Jeollanam-do) province. It's more countryside, but you can see the tea fields in Boseong, the bamboo groves in Damyang, eat seafood and southern style meals in Yeosu, and see the eco park in Suncheon! Jeollanam-do has a special place in my heart since it's my home province.
This is an itinerary I made for my parents. We had a home base in Seoul, but we also did a few days in Jeju. Two nights is definitely enough time. If you want, you can store your luggage at Incheon Airport and travel light to Jeju and then return to the airport just in time for your flight!
-- Agree to go Gyeonju.
-:- Message from TripAdvisor staff -:-
TripAdvisor staff removed this post because it did not meet TripAdvisor's forum posting guidelines with prohibiting self-promotional advertising or solicitation.
We ask all of our members to keep their forum messages free of self-promoting advertisements or solicitation of any kind - members affiliated with any tourism-related business should not include commercial contact information or URLs in their forum messages.
To review the TripAdvisor Forums Posting Guidelines, please follow this link: http://www.tripadvisor.com/pages/forums_posting_guidelines.html
We remove posts that do not follow our posting guidelines, and we reserve the right to remove any post for any reason.