Kremer – the Vilna Gaon had no Surname
(Portrait courtesy of Yeshayahu Winograd, Jerusalem)
Despite a common misconception, the Gaon had no surname and "Kremer" or “Kramer” was a nickname applied to his ancestor Rabbi Moshe "Kremer" of Vilna because he had a stall in the market.
The term "Kremer" means a shopkeeper.
None of the primary sources or documents contemporaneous with the Gaon and the generation after him include a surname, although several biographies in modern times have erroneously used it.
There are many, many Kremers/Kramers who have nothing to do with the Gaon.
However the name was adopted by the descendants of one of the Gaon’s brothers Rabbi Moshe of Podzelva. They lived mainly in Dokshitz, Belarus, and in Israel.
Another family descended is from a female connection to the Gaon, a daughter of his grandson Rabbi Tuviah Yurbarsky and wife of Rabbi Yitskhak Kremer of Volkovisk (there are descendants in the USA).
Unless a family has a specific tradition of descent from the Gaon, the surname Kremer alone is probably not sufficient evidence of a relationship with the Gaon.