I’ve done that before for my daughter.
1. Send an application for a new passort to a nearest Korean Consulate ( You have to explain in a document why a new name is required, e.g. to avoid harassment, etc.)
2. Receive a new passport with a new name
3. Send a copy of the passport to INS to apply for a new greencard
4. Send a copy of the passport to a Social Security Office to get a new social security card
5. Call a school district office to ask for a name change

This applies only to a minor. For an adult, I think the process may be more complicated due to some real estate issue, which I’m not sure about.