How would I go about creating a background texture that works like the default one, where it somehow looks good no matter how large you make the button?
The default texture isn't listed in the texture picker, so I can't see what it actually looks like.
To address this specific question. You can set the border as fixed by giving it a pixel size in the gui skin (about the pixel width of your boarder in the image). It will use that setting to stretch your custom background image in a specific way to prevent blurry/pixelated/crappy button backgrounds. Now you should be able to make a small image and have it show up correctly at any size.
I don't really see what background image you are talking about :p But if it is an example scene, you are probably talking about skybox, you can assign that in render settings (http://unity3d.com/support/documentation/Components/class-RenderSettings.html) or in your camera settings !