To add onto Delta's answer, the Black 2 version is incredibly similar to the White 2 version, and most of the version differences are based on which form of the main legendary Pokémon you will obtain, a few aesthetic changes to certain locations within the game, as well as the selection of Pokémon you will and won't be able to catch (both are incentives for trading with other people that have the other version). This means that the general plot and experiences with the game will be very similar, so playing through both is like playing a different mode of the same game. So yeah, there's no need to play Black 2 if White 2 was already played. :)
As for trying out the prequels to these games, though, that's a different story. Black/White is quite different from Black 2/White 2 despite the same setting. Still, it's very similar, so if your child starts looking for a different experience or setting altogether, then it might be better to try out Platinum or one of the HeartGold/SoulSilver titles instead.
Hope this helps.