vi is an interactive text editor included with traditional UNIX systems. It is the de facto (and de jure, by POSIX) standard editor for such systems.
For various reasons (e.g. AsimovAndTheCritic) there might not be any one "correct" pronunciation of vi. The "Vy" pronunciation gets reinvented constantly by folks outside the oral tradition. Some might comment that this is a salient point about what you name your software.
The original pronunciation, for what it's worth, as pronounced by ViEditor creator BillJoy, and everyone else at Berkeley, was "VEE EYE". I can personally attest to this, as I peered over his shoulder, kibitzed, and beta-tested his editors from the beginning (when he was fiddling with variants of ed) until he gave up on ViEditor development circa 1979. -- DougMerritt
Outside the oral tradition, one hears other stories:
In nearly ten years in the industry, working in the UK and US, I don't recall anyone ever pronouncing it "vee-eye". It's always been "vy" (to rhyme with "pie").
Many vi clones exist, many of which are briefly documented at http://www.saki.com.au/mirror/vi/clones.php3. Here are a few:
Mmmmm... One of my fondest memories is after hacking into an old dec at the univ. for the first time. I'm sitting there with a Vi quick-reference card trying to learn Vi fast enough to edit myself out of the super-user log and not get noticed by the administrator. I'm holding down x to delete each character in the line until I suddenly see the "dd" command - AhHa! I gave up hacking, but never Vi. I tried the same using emacs but without the same motivation it just wasn't meant to be...