I think people are too quick to assume things when it comes to the "i". Sure Apple popularized it with its behemoth known as iPod. Does that mean Nintendo wants to cash in on that? I'd go with "no" seeing as how every console name has been Nintendos' own, and not a copy of anything else. Go back a few years to when the term "walkman" was in, and even then Nintendo didn't go with "Game Man".
Whatever meaning Nintendo has with their product names is their own way of trying to be innovative in every aspect. The name "Wii" signifies the concept of "we" as in bringing together gamers to play, and also goes with "ii" instead of an "e" to represent two gamers playing.
The "i" on DSi can't even be compared to what Apple has done for a few reasons. First off, as mentioned in a different post, it's not called "iDS", and it's also not spelled "DSi". DSi is just how we spell it for simplicities sake, but if you take a good look at the placement, the "i" is where the ™ would be on a product name. On top of that, the "i" is in-cased in a black circle- resembling the very cameras that are in the DSi.
The final reason is this: Go to the roundtable discussion on how the DSi came to be, and you will see that they clearly wanted to give the DS "eyes", which is also an indicator that Nintendo is simply using the letter "i" to promote the biggest new feature to the DS, its "eyes"- in other words its new cameras.
I don't post much, but I will appreciate any feedback on the posts I do make. Feel free to PM