They're all pretty much the same, DD2 has had each of the DS' and has the new one on her list for christmas.
The DSi is where you get the camera with it - fun for taking pictures and making them funny (like squeezed faces, silly hats etc.) however, if your DS is only 6 I doubt he'll know the difference.
The screens get a bit bigger each version, DSLite is a bit bigger than the original DS, however as the name suggests, a bit lighter too!
DSi you get the camera, bit bigger screen, not much lighter tbh.
Mario games for all ages, there are the Ninten-Dogs/Cats/Whatever new ones they have come out with, which might be fun if your DS likes animals.
DD1 has recently picked up the Art Academy game, though it's not much of a game iykwim.
Teaches her to draw like.
There are the Big Brain games, which are fun, but you might want to check the age rating, whilst the rest of the family might enjoy it, DS might not :/. Though if he's into maths and stuff he'll probably enjoy it.
The DS's also fit the older games from the Nintendo gameboys, so you can get some classics like Sonic, but there are DS games for Sonic now too, a bit reinvented :).