I was trying to find out for a related post, but my googling skills failed me. All I know is that in HP Lovecraft's original work it was called a Sanatorium.
But when was the asylum first introduced to Batman fiction (Comic or otherwise)?
Arkham was introduced in the October 1974 issue of Batman, per Wikipedia:
Arkham Asylum first appeared in October 1974 in Batman #258, written by Dennis O'Neil and drawn by Irv Novick. In this story, it is named as "Arkham Hospital", although it is not clear what kind of hospital it is. "Arkham Asylum" first appeared in another O'Neil story the following year, but it was not until 1979 that "Arkham Asylum" completely replaced "Arkham Hospital", and the occasional "Arkham Sanitarium", as the institution's name. Also in 1979, the move to have the asylum closer to Gotham had begun; that was completed in 1980, when Batman #326 by Len Wein described the asylum's location "deep in the suburbs of Gotham City". It is perhaps for this reason that Batman #326 is listed in some histories as the first appearance of Arkham Asylum. It was also Wein who, in 1985's Who's Who #1, created its current backstory.