The YHA (Youth Hostel, is on the HI Hosteling site) next to Nam-Sam park is amazing, great facilities and cheap. About $17 AUS when I stayed per night and included a new towel each day, free fast WIFI, awesome location (next to the N Seoul Tower and Nam-Sam park which is really cool) and near Myong Dong (the main shopping district. I love Seoul and the YHA there made it even cooler. You can get a bus to near the hostel from Incheon International Airport for about $10 AUS which is easy and convenient. Hope that helps! about 8 years ago
There is another YHA in Seoul ; Songpa Gu, Olympic park, south of Han River.
If available, YHA is the best choice for foreigners.
And, there are a lot of Motels!
Um... Another available choice is... Ibis Ambassodor Hotel in Samsung dong, Gangnam Gu. about 8 years ago