If your mother stayed in Korea for one year or longer in one continuous trip, your mother’s “green card” (permanent resident status) is invalidated and it will be the Dept. of Homeland Security’s (DHS) position that she abandoned her residency here. However, even if her one continuous stay did not exceed one year, the DHS can still find that she had abandoned her residency.
In either case, she will have to go through the “Returning Resident” application (Form DS-117) and she will have to convince the DHS that her lengthy stay in Korea was for reasons BEYOND HER CONTROL and that she always had the intent to return and reside here in the U.S.
The document she was requested to submit is part of her such application. The key factor here is whether the lengthy stay in Korea was UNINTENDED for reasons BEYOND HER CONTROL. Evidence will be medical records for the treatment she received during her stay in Korea, while maintaining her residence here in the U.S. by providing utility bills, housing, driver’s license, car registration, financial/tax records, insurances, etc.
If she cannot provide documents to the DHS satisfaction, she will lose her permanent resident status. However, that does not prohibit her from applying for a brand new “green card” from scratch as a mother of U.S. child, or spouse.