Personally, they didnt really find a selling point for the system and attempted several ways and sometimes did it really half-assed.
I thought the Gamecube would've been a strong LAN-enabled system, but only three games ever supported the feature. Ironically all three games were racing games. Then they never bothered with such a feature ever again. Not even for any First Person Shooter games, it would've really helped the Timesplitters games definately. Their library had a similar problem the Nintendo 64 had, a good number of Nintendo's games were 4 player multiplayer games and a number of them were RACING titles. Wave Race, 1080, Mario Kart, Kirby F-Zero... They also didnt bother controlling the amount of Mario spinoffs that made it to the system. There was Mario Sunshine which people found 'okay' at best. Then there was Paper Mario 2 which i consider weaker (Not the weakest) Paper Mario game simply because it borrowed too many mechanics from the first game. (Partners had similar or the same abilities, combat was essentially the same) the rest range from sports, to DDR, to racing. Not exactly system seller material there.
Gamecube to Game Boy Advance connectivity was something they were advertising around the beginning of the GBA's life. It was soon around Pac-Man VS. that they really tried to hammer it down. Animal Crossing was the biggest of them all. Sporting not only the isolated island feature or sending NES games to the GBA, there was the e-Reader that let you get pieces of furniture, T-shirt designs and more. But that was it, all third parties could really do was unlock extra levels, basically forcing consumers to shell out cash for more content through these unimaginative ways. Not to mention that cord was really finicky, barely nudging the cord causes it to disconnect. The GCN to GBA cable and the e-Reader was also a small part of what Nintendo tried to expand the GCN life with: Perhiperals. The only way to play AND enjoy Donkey Kong Jungle Beat were to purchase a pair of Bongos. You could buy the game by itself for $40, which is really stupid. Since bongos cost about $30.
Its worth nothing that their blue water strategy did come in around the time Reggie came along advertising the idea of people coming together, playing games and enjoying the experience together. Nintendo had game catalog books at certain retailers and had pictures of people huddled at the TV together playing the Gamecube. But they still didnt really have the game that could help promote the idea.
This is partially why the Wii is a refinement. The Wii remote can be placed into cheap sometimes in-expensive plastic molds, you can connect the DS without any extra wires, though nobody is really jumping on this wagon still. Which provides an easy way to play that anybody can come in, which makes their advertising successful. They actually have some games that they can advertise as the party system without having to rely on their own properties that could alienate their hardcore userbase. So Super Paper Mario, even though based on an RPG series, is a platformer thats very easy to get into.
The only problem with the Wii is obviously the third party support. They feel that the only thing that sells on Wii are party games. But since most developers just churn out a product without actually putting any real effort into it, it just flounders. They basically offer the same thing Nintendo already did, like boxing or skiing and its worse because Nintendo already pulled the idea off. Nintendo also has a better idea how their hardware works, so 3rd party developers can usually get it wrong and basically not work properly, like in Monkey Ball. So they want to emphasize how to play games by saying (Well you have to stand like this and keep one leg up, otherwise it doesnt work.)