I remember the days before Nintendo totally ****ed up GC-GBA connectivity.
One great idea for it was continuing the same game on both machines, like playing Oblivion on PS3 and enter a special dungeon and play on in the PSP version, which was/is planned as a dungeon crawler, as far as I understood. The key point being the same character being transferred and levelled up in both versions and with maybe even the stories could be connected.
I think this is still a nice idea, unfortunately it needs a lot of care put into 'syncing' the game design.
But now we have remote play and I had a slightly different idea. The main problems of remote play are lack of buttons on PSP as well as possible lag or unreadable texts. I dont know if Sony is already thinking in that direction, but the solution is 'easy', just give the game an extra remote play mode.
That mode might have the buttons configuration simplified, maybe has the difficulty reduced to make up for the 'worse' control sheme. Same might go for inventories/selection screens, which would be optimized for the PSP screen with bigger text or icons.
A more drastic, but of course possible approach would be lets say a realtime PS3 game that transfroms into a turn-based game when in 'PSP mode'. Anyway, all this would result into one continous game experience, just leave your PS3 on when you leave the house (or turn it on again remotely) and continue the game exactly where you left off, just on your PSP.