Apple has been using a lowercase "i" to begin many of its product names ever since Steve Jobs introduced the first iMac computer on May 7, 1998. His explanation for the "i" at the time was as follows:
iMac comes from the marriage of the excitement of the Internet with the simplicity of Macintosh. Even though this is a full-blooded Macintosh, we are targeting this for the #1 use that consumers tell us they want a computer for, which is to get on the Internet -- simply, and fast. And that is what this product is targeted for.
"i" also means some other things to us. We are a personal computer company, and although this product is born to network, it also is a beautiful stand-alone product. We are targeting it also for education. They want to buy these. And it is perfect for most of the things they do in instruction. It is perfect for finding tremendous sources of information over the Internet. And we hope as you see the product it will inspire us all to make even better products in the future.
Internet. Individual. Instruct. Inform. Inspire. Steve Jobs may have originally used those words to describe the iMac, but they sure do apply quite nicely to the iPhone, don't they?
Of course, there was a long road from the "i" in iMac to the "i" in iPhone. In July of 1999, Apple announced the iBook, Apple's consumer laptop (which evolved into the current MacBook). In October of 1999, Apple announced the iMac DV and introduced iMovie, Apple's consumer video software. On January 5, 2000, Apple released iTools (which evolved into the current MobileMe), iCards (now defunct) and iDisk. On January 9, 2001, Apple released iTunes and iDVD.
Later that year, on October 23, 2001, Apple introduced the iPod. In Steven Levy's book about the history of the iPod The Perfect Thing -- a great book that you should buy immediately if you would enjoy a behind the scenes look at Apple -- he discussed the origin of the term "iPod" as follows:
It was Jobs who told everyone what the device would be called. "He just came in and went 'iPod,'" says one team member. "We all looked around the room, and that was it. iPod. And we're like, 'Where did that come from?'" (Excellent question, and one that proved increasingly elusive the more I pressed people at Apple about it. Finally, I was able to corner Jobs on it and he said that to the best of his knowledge the name sort of emerged, not exactly in a form of immaculate conception but in a lengthy back and forth among him, his marketing people, and Chiat\Day. "The ad agency loved it," he told me. But I got the distinct impression that the iPod moniker won out not because of its brilliance but because Jobs had had enough of the naming process and the hour was getting late.) [pp. 46-47]
[UPDATE 11/29/11: In the biography of Steve Jobs by Walter Isaacson, there is a single sentence describing the origin of the name: "One of the copywriters suggested they call it a 'Pod.' Jobs was the one who, borrowing from the iMac and iTunes names, modified that to iPod."]
After the iPod, other Apple "i" software products followed such as iPhoto and iChat in 2002, iLife in 2003, and iWeb in 2006.
Considering Apple's long history of "i" products, it may now seem that it was obvious what Apple would call its phone. But it wasn't always clear that the iPhone would be the iPhone. Before January of 2007, there were rumors that Apple was working on a phone and speculation over what Apple might call it. At the time, "iPhone" seemed like the logical front runner, but other names were rumored such as "iChat Mobile." (Yuck!)
Even after Apple announced in January of 2007 that it would ship the iPhone later that year, there was still some uncertainty about the name because Cisco insisted that it acquired the trademark to "iPhone" in 2000 when Cisco bought a company called Infogear Technology. Just after Apple announced the iPhone, Cisco sued Apple. Apple responded that other companies besides Cisco were using the term "iPhone" and that Apple was the first to use "iPhone" to refer to a cellphone. (I doubt that Apple would still take the position today that anyone can use "iPhone" for a product.) At the time, there was even recent precedent for Apple changing a product name to drop the "i" -- on September 12, 2006, Apple announced a product that it called "iTV," but then at the same Macworld Expo at which the iPhone was announced, Apple said that it would call the product Apple TV (presumably because Elgato was already selling a product called EyeTV.) With all of this uncertainty, the first (and unfortunately, now gone) iPhone podcast called itself the "Apple Phone Show" instead of the "iPhone Show" in part because the show's creator, Scott Bourne, wasn't sure that Apple would use the name "iPhone" when the product shipped.
Eventually, of course, the uncertainty ended. The Cisco-Apple lawsuit settled on February 21, 2007. The terms were confidential, but Apple announced that the companies agreed that both companies could use "iPhone" and that they would explore opportunities to work together on issues like security and enterprise communications. (And sure enough, there is a Cisco VPN client on the iPhone.) By the time the iPhone was released on June 28, 2007, uncertainty about the name had already entered the history books.
So there you have it, a short history of the letter i. If you want to return to those halcyon days of youth when Apple first announced the "i," you can still watch a YouTube clip of Steve Jobs introducing the first iMac. This was soon after Steve Jobs returned to Apple, and he had not yet started exclusively wearing his now trademark black turtleneck and jeans for his announcements. But it was still vintage Steve Jobs, full of quips such as this one: the back of the iMac looks better than the front of everyone else's computers. Here is the clip, which runs seven minutes: