The girl's name is Walda Pepper. If your phrase makes her happy, you receive a custom wallpaper that you can use in your PC boxes! What the wallpaper looks like depends entirely on the phrase you give Walda, which is always unintelligible like "snhBCLpqgQBFVNT" or "jdhBPNdhnBnHVcf".
To illustrate, below are some examples of wallpapers you can make. You can pick from any combination of 15 different "base" patterns and 29 different "icons", plus any two RGB colours to give it a personal finish. Suffice to say, the below is barely scratching the surface.
Go to pokewiki.de's secret code generator and select "Emerald" to make your own design, then press "Calculate" to get the code which will create it in your game. Different people will have different codes for the same design because their Trainer ID is one of the factors used to compute wallpaper codes.
To get a different wallpaper, give Walda a different phrase. You can only have one custom wallpaper at a time, so make sure you save the codes for wallpapers you like. To set the wallpaper, go to your PC and set the selected box's wallpaper to "FRIENDS" (which is always the name of Walda's wallpaper).
It seems like this feature was intended to be used by official sources to distribute special wallpaper codes, but this only ever happened once on the Japanese Pokemon Fan Club's website in 2004, which handed out this wallpaper. Fans later reverse engineered the Walda algorithm to discover the full functionality!
