Everyone calls the forthcoming iPhone the iPhone 5. Not everyone, but that's generally what its referred to as when any rumors or speculation about the device are reported.
In reality, it's very unlikely that the device will actually be called the iPhone 5. Remember the iPad 3? That was what everyone was calling the latest version fo the iPad before Apple unveiled it, and simply called it the "New iPad".
I'm not really sure how helpful the strategy of naming a device the "new" iPad or iPhone really is, because eventually, it will no longer be called the "new" one, but that is clearly the path Apple has taken with the iPad, and it seems only logical that the iPhone would follow a similar path.
Gizmodo makes a pretty good case for why the device will almost certainly not be called the iPhone 5, beyond the above point. As the publication notes, the original 2G iPhone was just called the iPhone, the second one was called the iPhone 3G, the third one was called the 3GS, the fourth one was called the iPhone 4, and the fifth one was called the 4S. The major point being that the fifth generation iPhone already exists, and it's not called the iPhone 5 (remember, that's what everyone was calling that one too before it was unveiled).
It could be the iPhone 6, but that seem fairly unlikely as well.
If I was a gambling man, I'd put my money on "New iPhone" as what Apple will actually call this thing. Apple still refers to the latest iPad as "The New iPad". You can see it that on the company's iOS 6 page, where it shows you the devices the operating system will be compatible with.
It's not hard to imagine "The new iPhone" being added to that image. Or maybe it will be "iPhone 6th generation".
What do you think?