A tough one actually. The gamer in me would suggest getting a 3DS - as they will certainly be future-proof as a games system. The DS as a system is now 7 years old almost and while Nintendo are still getting great sales on the older DS models - they will phase out and discontinue the system in the next year or two (as is the habit with video game console cycles)
But then we are not talking about a dedicated gamer, but a kid - kids don't always need the latest tech and the DS has a fine back catalogue of games that any kid will have a ball with. On that basis I would go out with the intention of getting a DS Lite model where you get the best of both worlds - as the Lite will take DS AND Game Boy Advance games - many of which can be picked up for super cheap prices. So it broadens the range of games you can purchase for the thing...
Though on the flipside, the DSi - while locked down to just DS cards - has access to an online store where you can download DSiware - cheaper games that you download and play. The 3DS takes that further with a "Virtual Console" section in it's online store - selling classic Nintendo Game Boy, NES and Sega Game Gear titles for pretty decent prices.
Essentially it comes down to how keen your kid is about gaming, and what your willing to pay - and how long you expect the system to last before they want the next model up. I think if I were you...I would get a cheap preowned DSLite and some games, and see if she gets into the habit much or not - then if she asks for a 3DS or something, get that a little later on (shes a year too young for the recommended age to make any use of the glasses-free 3D anyway)
If you go preowned, always inspect the system before purchasing, the only problem with DSLite systems is the hinges are a little prone to breaking when too much pressure is applied to opening the clamshell design. Also check for scratches in the screen, it's quite possible to get a reasonably well looked after system in the preowned market. But never go in blind with these things, a scratched touchscreen is no fun. Also get screen protectors and apply them asap, they'll help the system last a lot longer (as the most common issue with any DS system, is scratches on the touchscreen from too much pressure applied to the stylus.)
This. We got our DS a DS Lite last year, for his 7th birthday, while he does play it quite a bit, I am wishing we had just sprung for the iTouch instead. The iTouch is $200, but the games (apps) are SO much cheaper. DS games are $20-$40 EACH, so that adds up quickly. Besides, no cartridges with the iTouch. We are considering either iTouches or Kindle Fires for our boys' next birthdays. --Katie
Comparing iOS app games to dedicated handheld games is like comparing Oranges to Apples for the most part. I have both platforms and I LOVE my iPod Touch, but outside retro games - those 99 cent apps are cheap because they operate on limited assets and quick turnaround, so most of those cheap games, while somewhat addictive - do not have any real sufficient length to keep someone going for hours. Where as DS games have longer development time, and has more unique content to offer per game. Plus the fact the iPod touch is strictly limited to the touch screen for controls...games like Angry Birds work a charm - but anything requiring more precise controls (like Platformers - Your Mario's and your Sonic's) can never hope to be as good as the handheld experience because the touch controls are kind of fiddly.