Dear lord. Dat list is gonna be so long.
It's not a DSi, though, right? Because if it's a DS(Lite), we could also throw in GBA recommendations lol.
Alright, let's go:
Action
Contra 4
Mega Man Zero Collection
Mega Man ZX / ZX Advent
the three Castlevania games
Aliens: Infestation
Ninja Gaiden: Dragon Sword
Star Fox Command
Action Adventure
Scurge: Hive
Zelda: Phantom Hourglass / Spirit Tracks
Puzzle/Platformer
Henry Hatsworth in the Puzzling Adventure
Kirby Mass Attack
N+
RPG
Contact
Chrono Trigger
Radiant Historia
Dragon Quest V, IX (feel free to also get IV and VI)
The World Ends With You
Mario & Luigi: Bowser's Inside Story
Infinite Space
Solatorobo: The Red Hunter
Strategy:
Fire Emblem: Shadow Dragon
Advance Wars: Dual Strike
Ninja Town
Lock's Quest
Visual Novel/Puzzle/Adventure
999 - 9 Persons 9 Hours 9 Doors
Ghost Trick
Rhythm
Rhythm Heaven
Elite Beat Agents
Jam with the Band
Puzzle
Meteos
Might & Magic: Clash of Heroes
Soul Bubbles
Tetris DS
Picross 3D
Other
Retro Game Challenge
Kirby Canvas Curse (and the Power Paintbrush in EU)
GTA Chinatown Wars
Geometry Wars Galaxies