The traditional answer is rice and soup. The modern answer, for a lot of people, is coffee on the way to work.
A Korean breakfast is structurally the same as lunch or dinner: rice, a soup or stew, and several banchan. There is no separate category of breakfast food the way there is in the West. Gukbap — rice in a hot soup — is the classic morning dish, and gukbap restaurants are among the few places reliably open at 7am.
Surveys consistently show a large share of urban Koreans skip breakfast or replace it with coffee and a pastry. Toast stands near subway stations and university gates sell egg-and-cabbage toasties, and convenience stores sell triangle gimbap for the same purpose. This is now the more common weekday reality.
Gukbap and haejang-guk (hangover soup) restaurants often run 24 hours or open before dawn, because they serve night-shift workers and people leaving bars. Traditional markets start early. Most other restaurants — including almost all grill houses — do not open until 11am, and cafés typically open between 8 and 10.
Hotel breakfast buffets in Korea usually offer a Korean corner alongside a Western one. If you want the local experience, a gukbap shop near your accommodation is a better use of a morning, and costs roughly the price of a coffee at a hotel.