It looks like it's possible that, after the first load, the banner information is stored on the Wii's NAND memory, which means, for modding it in Dolphin, you'd have to go to your "[Dolphin Directory]\Wii\title\00010000\[Game ID]\data" folder, to find the banner.bin that is loaded when the disc is put in (and I think this would also change the banner/icon that is used in Dolphin's game list). Search for "Wii Banner Creation Guide" on GBAtemp, and you'll find a thread that has very detailed instructions on how to edit the icon and the banner, including images, animation, etc, as well as links to the tools needed to get the banner to an editable state.
You can find the [Game ID] by right-clicking the game in your game list, clicking Properties, going to the Info tab, and looking at the "Game ID" field under "ISO Details". The number you're looking for to use as the [Game ID] folder name is the number that starts after the last 0 in that huge number in parentheses (so, for example, the name of the folder for Rune Factory Frontier, which has a value of "0001000052554645" in the Game ID field, would be 52554645).
Unfortunately, it looks like that would only work for the Wii games. In order to do the GC games in the Wii Menu, I think you'd have to edit the "opening.bnr" file in the ISO itself, and I couldn't find any information on extracting that, editing it, or putting it back in the ISO.
You can find the [Game ID] by right-clicking the game in your game list, clicking Properties, going to the Info tab, and looking at the "Game ID" field under "ISO Details". The number you're looking for to use as the [Game ID] folder name is the number that starts after the last 0 in that huge number in parentheses (so, for example, the name of the folder for Rune Factory Frontier, which has a value of "0001000052554645" in the Game ID field, would be 52554645).
Unfortunately, it looks like that would only work for the Wii games. In order to do the GC games in the Wii Menu, I think you'd have to edit the "opening.bnr" file in the ISO itself, and I couldn't find any information on extracting that, editing it, or putting it back in the ISO.