The second part of Iwata Asks DSi discusses the possible features that could have been implemented.
When considering the device's functionality, Nintendo felt the need to strike a balance. It is a new piece of kit, but still had to retain the DS's core functionality and design, a sense of newness. Some of the ideas from included the ability to do video calls, even an all-purpose image editor but only two or three of these could be implemented in time for release. A few can be downloaded at a later stage via the DSi Shop (much like the Wii's add-ons like Browser, News etc).
A lot of work went into developing the user interface. It had to be clear that photos can be taken, set as wallpaper, effects added etc, by easy to read menus and icons. Given the handheld nature, the DSi needed to be more of a "pick up and play" device, easy to get into but with depth for those wanting to take it further.
Another interesting point is competition. Nintendo don't have any intention of competing with existing products (iPods and mobile phones) by adding in cameras and a music player, supposedly. Iwata feels it's due to a need to "pleasing as many people as possible who pick up a DS" and hopes those who think this is out of character for Nintendo would pick up a DSi themselves.
More on the design and functionality in Iwata Asks DSi - Volume 2.