You can use the Mystery Gift feature in Pokémon Brilliant Diamond and Shining Pearl in order to receive various promotional items and Pokémon, such as the Manaphy Egg that are given to everyone who buys the game before February 21.
However, you do need to unlock the Mystery Gift feature in-game before you can get any of these, and you have to do a very specific thing to do it. Here's how you unlock the Mystery Gift:
To unlock the Mystery Gift feature in Pokémon Brilliant Diamond and Shining Pearl, head to Jubilife City after having gotten the first gym badge. Exit through the north exit and fight against Team Galactic. Now go to the Jubilife TV building (used to be blocked by a clown) and go to the third floor. Talk to the left of the the persons in the bottom of the room and answer their questions with "Yes." first and then "Everyone happy wi-fi connetion".
Once you do that, the Mystery Gift option will unlock for your save file. To use it, open the meny, choose the Mystery Gift there, and then "Get via Internet" if you want to get your Manaphy Egg.
When you have unlocked it, the Mystery Gift feature is available forever on that save, so you do not need to go to the Jubilife TV building to use it in the future.
The Mystery Gift feature will unlock naturally after beating the third gym
If you don't bother with method above, the Mystery Gift feature will naturally unlock after you beat the third gym leader in the game. This is just so that people who don't either follow a guide or stumbles upon the woman in the Jubilife TV Building also gets to use the feature.