From the Super Mario Wiki
Glowing Rabbits are rare, shiny types of rabbits that appear only in Super Mario 64 DS. These rabbits appear to be white with sparkles appearing from their body. When 50 Power Stars are collected, Glowing Rabbits start to randomly appear around the castle in place of the character's other rabbits, but they are so scarce, it's rare to find one. If the player uses Yoshi, he can catch six of his rabbits outside in front of the castle. He can also get the other two if the player goes back in and outside the castle repeatedly until the final two glowing rabbits appear. There are eight Glowing Rabbits total. When all eight rabbits are caught, the player is rewarded with a key that unlocks the last door in the character selection room, which contains a secret Power Star. Glowing Rabbits no longer appear after finding all of them.
The final Glowing Rabbit caught states that it is the eldest of the Glowing Rabbits. Additionally, they appear to be related to each other and are all male, as each rabbit then before the last one states that it is the "second fastest of [their] eight brothers."
If a rabbit before the final eight one is caught, the player can get a chance to save their current progress.