Let me introduce you Seoul's district for young people at heart
Hyehwa [or 혜화 in Korean] is located in the north eastern part of central Seoul and has its own subway station (see map here). My hostel was located nearby, withing 10 min of walk. I went here every night and had something to eat. The food is really delicious here, but the prices may not be low. The people here mostly young, you can see many neatly dressed guys and some very hot girls. Maybe Hyehwa can be a place, where you would bring your girl on a first date? Who knows. It's definitely a very neat, clean and fun part of Seoul and there's something going on every day, be it theaters, pubs or restaurants. Open all the time. Enjoy some photos.