I believe a Gamecube handheld is definitely not part of Nintendo's plans for now... It would be just too expensive and Nintendo never releases a very expensive console...
If PSP really comes with all the technology announced it's not coming with the price range of a GBA... A PDA with such technology costs at least $400. Surely, this technology is going to be cheaper one year from now when PSP is going to be released, but, anyway, it's not gonna be cheap.
I'm afraid Nintendo is not bringing many improvements to the next Gameboy for such reason$. I'd love to have a gameboy with a new form of media, better screen, stronger processing and more buttons, but I don't know if it's going to come true...
If Nintendo gets rid of the optical discs issues (batteries, loading times and durability) I guess the best media would be a case-protected optical disc (like PSP). Memory sticks would also be nice, but they're still too slow for such applications and very expensive for now.
The main issue about changing media is the loss of backward-compatibility (and I'm almost sure Nintendo won't let it happen). Unfortunately, I agree that it's gonna be tough for the Gameboy to face PSP without some technological evolutions... Let's hope Nintendo hears us all!!!