Lane tells Rory her “Korean name” – it is unclear whether this is her legal name, and Lane is a name to be used among English-speakers, or whether Lane is her legal name, and her Korean name is a middle name given to help preserve her culture, and to be used when she is among people of Korean heritage. I suspect the latter.
The teletext says that Lane’s Korean name is Hyung-Hyung, which is highly improbable. Hyung is not a Korean name, but a title of respect given to address a male, literally meaning “older brother”. The Kims would have been completely nutty to choose that as Lane’s Korean name, and if done deliberately, must be a joke by the scriptwriter (Amy Sherman-Palladino).
It is possible that Lane actually says her Korean name is Hyun-Kyung, which can be translated as “virtuous respect”. It’s a reasonably common Korean name for girls, and there are several famous women with the name.
(Lane seems to have bought two lava lamps from Andrew’s bookstore, Stars Hollow Books).
The Annotated Gilmore Girls 