Will also be in Seoul from Oct. 15-21, I frequent the city 2-3x yearly to visit family and my dentist. What part of town are you staying?
Some of the more famous food places I've tried and like:
1. Tosokchon Samgyetang - I really like the black chicken version of their signature soup dish, and the kimchi is quite good too. This is nearby Gyeongbokgung in Jongno and makes a great meal before or after exploring the palace.
2. Pyeongando Jokbaljip - If you are into pig trotters, this place is one of the oldest in the city and my personal favorite. The flavor is less sweet and more "developed" so to speak. Nearby Dongkuk University.
3. Noryangjin Fish Market - there is an "old" open air part of the market and a "new" indoor multistory part. I prefer the old market, more charm and the cooking is better. You can pick whole live fish and have it as sashimi on the spot, they will also boil the head and bones into a spicy soup which is delicious. The fishmonger will refer you to "their" restaurant, usually a partner business who does the cooking side only. Can go any time of day but as with fish markets in general, mornings are better.
Recommend shopping around and comparing prices between stalls, they all carry mostly the same product but the prices can vary a lot and there is always room for negotiation. No shame in getting a few prices and going back to the lowest price vendor, its normal. Don't wear sandals or nice shoes, they will get dirty and wet.
4. Ichi Ryu - Lamb BBQ specialist, "Genghis Khan" style. Near the "Hongdae" area, known for its street performers and active nightlife. Went here with my Uncle and it was delicious. It is popular and gets busy on the weekends. Go early in the evening or prepare to wait.
5. Vegetable "Ho Dduk" - Many places sell this street food item, but the best known and tastiest is in NamDaeMun Market, gate #2. Its a fried bread pocket with vegetables and noodles inside. Usually a long line but worth it!
Hope you enjoy Seoul and feel free to PM me for any additional info
-Pete