The PSP actually did remarkably well all things considered. Beating Nintendo at the handheld game was a longshot but Sony did manage to snag a respectable amount of marketshare. The PS3 suffered not because it had Playstation in the name, but because it was launched at $599 with its biggest title being a WW2/Alien FPS (which at the time was completely outclassed by Gears of War in hype). And now the Vita is suffering because of a relatively high price point and no hit games.
It's not rocket science. Instead of dropping the Playstation name, make a console with a reasonable price point, a variety of media features, and games that people want to play. Considering that the PS1 and PS2 were successful at a $299 launch price point, maybe that's a good target to aim for.
