The National Hangeul Museum near the National Museum of Korea tells the story of Korea's unique writing system: its invention by King Sejong in the fifteenth century, its suppression during the Japanese colonial period, and its development into the modern alphabet used by 80 million people. The permanent collection includes manuscript examples, printing blocks, and interactive displays explaining the phonological principles behind the alphabet's design.
The museum is free and the exhibits are well-designed with English captions. It's a good complement to a visit to the National Museum of Korea next door, and a useful place to understand something of Korean cultural distinctiveness. Allow about ninety minutes.