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", and varies depending on your in-game Trainer ID.
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.
To create your own wallpaper and generate a code for it that works in your game, go to this website...
...then select "Emerald" and type in your Trainer ID (seen on your trainer card in-game). Then, press "Calculate" to get the in-game code for the wallpaper you made. Different people will have different codes for the same design because your Trainer ID is one of the factors used to compute wallpaper codes.
To get a different wallpaper, generate a different code and give it to Walda. You can only have one custom wallpaper at a time, so it will replace your previous one. Write down codes for wallpapers you like if you want to get them back later.
To set the wallpaper in your PC boxes, go to the 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!
