First off, the Kingdom Hearts story is presented in a chronological order. I have bolded the ones that can be played on your daughter's DSi.
BBS-Prequel to the first, Play Station Portable system
Kingdom Hearts I-Play Station 2
Chain of Memories-Game Boy Advanced. Can only be played on first generation DS's, but not on DSi, because they don't host a game boy advanced slot.
358/2 days-DS game. Can be played on your daughter's DS.
Kingdom Hearts II-Play Station 2
Re: Coded-DS game. Can be played on your daughter's DS
I recommend that she either watch, play, or read up on the first game, as it might help explain the plot of the 358/2 game more (which features Roxas, a character that is prominent in the second game, and that we don't even meet in the first game).
Re: Coded has the original characters, Sora/Mickey/Donald/Goofy, but it takes place after Kingdom Hearts II.
I thought that story wise, 358/2 days added to the series more, and was a better/smoother game to play overall than coded. It does not, however, have Sora/Mickey/Donald/Goofy featured in it.
Coded introduces a newbie to the main characters, and goes through the "worlds" of the first game, but it can be complicated to get used to as I found some of the controls a bit funky. It also sets up the beginning story for the next installment in the series, which will be on 3DS (Dream, Drop, Distance).
For a newbie, I'd probably recommend coded, because the story is somewhat easier to understand than 358/2 days, and it's the closest in plot to the first game.
