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).