During our weekend excursion we stayed at the Lavinia hostel near Anguk (안국) station, in Seoul. It was really clean and the facilities looked pretty new. The staff was very kind and helpful (we were allowed to keep our bags there after we checked out so we could go shopping and hang out without the burden of our backpacks. It also includes a complimentary breakfast every morning. Breakfast is simple but enough and it’s one less meal that you have to pay for while travelling.
The bottom few floors of the building was more of a backpackers hostel. I think it was called Banana Backpackers. In case you’re ever looking for something in the area and Lavinia is full.