The Unix section of the wiki page on configuration files, leads to the wiki page on run commands, which is where the rc comes from. Quoting from the run commands page:
In the context of Unix-like systems, the term rc stands for the phrase "run commands". It is used for any file that contains startup information for a command. It is believed to have originated somewhere in 1965 from a runcom facility from the MIT Compatible Time-Sharing System (CTSS).
Additionally, they refer to vim with the following:
While not historically precise, rc may also be expanded as "run control", because an rc file controls how a program runs. For instance, the editor Vim looks for and reads the contents of the .vimrc file to determine its initial configuration. In The Art of Unix Programming, Eric S. Raymond consistently refers to rc files as "run-control" files.