Honggik University (also known as Hongdae) is an art and design university. Because of this, the neighborhood around it is filled with young people, musicians, and artists all looking for a place to be creative. This makes Hongdae the perfect place to sit in a cafe all day and study, read, or just relax.
Tora-B is a great place to start - you can sit on the floor, out on the patio, or at a table and enjoy the natural lighting and fresh coffee.
I've attached a map of how to get there, so head to Hongdae Station, exit 8 and follow the map :)
All photos from this blog: http://blog.naver.com/bbo0305/220187350099