What does Vi stand for?

Wikipedia has a concise and properly cited explanation:

The name vi is derived from the shortest unambiguous abbreviation for the command visual in ex; the command in question switches the line editor ex to visual mode.

http://en.wikipedia.org/wiki/Vi

Note that this does not mean that the full name of the text editor "vi" is still "visual" or, even, that "vi" currently stands for "visual" - the name of the editor is just "vi". The above quote just explains how the name originally came into being.

As Ben Voigt helpfully points out, the people (and websites) claiming that it stands for "Visual Instrument" are confusing it with another product unrelated to the text editor.

Vi is a screen editor for Linux, Unix and other Unix-like operating systems. Pronounced (vee-aye). Stands for VIsual.

_{Sources: http://en.wikipedia.org/wiki/Vi and http://www.techopedia.com/definition/10347/vi}

If one goes back and looks at an old man page for vi, such as the one for SunOS 5.10 from '07:

SunOS 5.10          Last change: 16 May 2007                    1

User Commands                                               vi(1)

     The vi (visual) utility is a  display-oriented  text  editor
     based on an underlying line editor ex. It is possible to use
     the command mode of ex from within vi and to use the command
     mode of vi from within ex. The visual commands are described
     on this manual page; how to set options (like  automatically
     numbering lines and automatically starting a new output line
     when you type carriage return) and all ex line  editor  com-
     mands are described on the ex(1) manual page.

Or from Ultrix (latest release '95):

Name
       vi - screen editor

Syntax
       vi [ -t tag ] [ +command ] [ -l ] [ -r ] [ -wn ] [ -x ] name...

Description
       The (visual) editor is a display-oriented text editor based on The com
       mand and the command run the same code.  You  can  access  the  command
       mode of from within

And thus you have it: vi is from vi‍sual. This matches the pattern its ancestor editors: ex is the ex‍tended editor and ed is an ed‍itor.

There is no indication that this stands for anything else. Maybe V‍isual I‍nterface (what I learned it as back long ago), but there is no documentation that it stands for anything other than vi‍sual.