When people search for the best area for plastic surgery in Seoul, they usually mean one district: Gangnam. It holds the highest concentration of cosmetic clinics anywhere in the world, packed into a handful of neighborhoods that each have their own personality. The big four are Gangnam Station, Apgujeong (and Apgujeong Rodeo), Sinsa / Garosu-gil, and Cheongdam.
You don't have to overthink it. If you want the widest range of clinics and price points, base yourself near Gangnam Station. If you want the polished, celebrity-doctor end of the market and don't mind paying for it, look at Apgujeong and Cheongdam. The whole belt is walkable and connected by subway, so wherever you book, the others are 10 to 20 minutes away. One quick legal note before we start: a clinic being government-registered to treat foreign patients is a licensing status, not a quality score. It means the clinic is set up to handle international patients properly. It doesn't rank one clinic above another.