Many Korean traditions date back centuries (some even to the B.C. era), and the use of the dojang is one of them. A dojang is a personal seal, also called a name chop, that is used to stamp important documents, either in lieu of or in addition to a signature.
In the past, the seals were used exclusively on important documents (no signature required). And while that’s changed some today, the seals are considered more formal. It is also more traditional for the name chops to have the Hanja characters (Chinese characters) on them, but again having hangul characters is becoming more common as well. 
These are great souvenirs to bring home, but I have to admit that we didn’t get our son a dojang while in Korea. I decided it might be more fun for him to be there when we had it made, and get to choose how he wanted his name done on the chop. So, it’s another thing to add to our “next time” list.
Shops that make name chops are numerous. But I’ve had a lot of people recommend a shop in Insadong. It’s the Myung Sin Dang Pilbang (Brush Store), located at 18 Gwan Hun Dong Jong Ro Gu, Seoul, Korea.
Here’s the name and address in hangul:
명신당필방
서울시 종로구 관훈동 18
02-736-2466

Photo by Mary Leigh
This shop is featured on the Korea Tourism page about Insadong. Here’s the link: http://english.visitkorea.or.kr/enu/SH/SH_EN_7_4.jsp?cid=561239. And here’s a photo of the front of the shop, which might help you in located it as well.