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}