Now, theoretically, I think this could be solved. Anyone remember xbox cpu mods on the original xbox? I wasn't really in the scene, but there are many whom have replaced the cpu with even stronger ones (of the same architecture of course). And I believe this should be possible with the NGC as well.
Let's start with the gamecube's processor, the Gekko.
It clocks at a maximum 486MHz, which is a very limited amount of power.
It's Microarchitecture is part of the PowerPC ISA 1.10, or the PowerPC G3.
Now let's take a look at the wii's processor, the Broadway.
It clocks at a maximum of 729MHz, which is quite the power kick compared to the Gekko (50%)
It also has the same architecture, the PowerPC ISA 1.10, or PowerPC G3.
It also consumed 20% less power than the Gekko, so power consumption would NOT be a problem.
It is also reported, but not confirmed it is directly derived from the Gekko (basically just an upgrade, but otherwise the same).
It is also indeed half the size of the Gekko, but it should still be possible to connect it to the GC regardless of size (with a few modifications).
Most importantly, is is noted to be Backward compatible with the Gekko processor, which is fantastic for these purposes.
Now even more interesting would be looking at the Wii U's processor, which is the Espresso.
The Espresso clocks at a way more powerful 1,24GHz (OVER TWICE THE NGC'S POWER!).
Now the most important parts:
It is indeed also part of the PowerPC ISA 1.10, same architechture still going (Well, nintendo has always just upgraded console by improving the previous, but you get my point).
Even so, it is BACKWARD COMPATIBLE with BOTH the Broadway AND the GEKKO (Wii U can actually play NGC games with a controller adapter and some slight modifications with softmodding).
You guys see the pattern I see? The Wii U proccessor is possibly compatible with the Nintendo Gamecube (I am not sure about heat issues, but those of us with Wiikey's/Wasp's/Wode's have good enough space to replace fan and heatsink to be more efficient).
Though, it is less likely to use the wii u proccessor on the gamecube than the wii on the gamecube as they appear more similar (virtually identical imo). Also the Wii U is three cores (so 3GHZ total, but i don't know if it's possible to use three cores on the gamecube) and not one.
The wii cpu (PowerPC 750CL, kinda. it is just a 750CL with a higher clock speed haha) is very likely that the 750CL is a shrink of the same processor for general purpose use (wii+gc).
I'll add a comparison right here:
IF anyone has any information not noted down here, please do help the cause.
My hopes aren't too high, but high enough to start researching the chips down to each and every damn pin.
I'd focus on the Wii more than the Wii U though, as it has higher potential for success.
tl;dr: They did it with the Xbox, Now I want to do it with the NGC and have reasons to believe it could work