For most tourists, it makes sense to stay in the Jongno area, to be easily accessible to most historical tourist attractions, as well as some of the main other areas to visit on a first trip.
As mentioned in the Key Areas (Part 1), since it's hard to pinpoint the "city center", that also means that there isn't really a premium for staying in any one area over another. There are nice hotels, Airbnbs, etc. all over the city, and Jongno is not a particularly expensive place to stay.
Looking at the subway map of Seoul, you could also easily decide to stay elsewhere that is well connected to Jongno. Hongdae, for example, is very well connected by line 2, so if you enjoy staying out late, it could make sense to stay in Hongdae and commute to the other areas every day. (Then, for example, one can enjoy the nightlife of Hongdae without worrying about catching the last train.)
