Ooookaay, having read the whole thread, mostly laughing at people for various reasons (mostly stupidity and stubborness...) and thinking about how I used to play Pokémon, comparing it with the types of games I play nowadays, I think it's my turn to post, eh?
Most important thing first: some kind of replay-value. If you have a split story line, and even different beginnings to choose from, you're already good to go. I'm not speeking of choosing a starter here, it's more like having the player answer the question "As who do you want to play?" more precisely than "A girl (if even possible)?". Take Dragon Age: Origins as an example: you can not only make your character look the way you want - which also is NOT what I'm talking about here - you choose a race (human, elve, or dwarve), a class (warrior, rogue or mage), and depending on that, you may choose from 1-3 different so-called origins (there's six in total, by the way). Let's say you - like me in my first playthrough - picked the elven rogue; You could the choose whether to start as an elf in the slums or as a so-called 'Dalish elf', basically an exiled elf, living in the forest, in piece with both nature and his clan (it's less clicheé than that, I'm just trying to have you guys understand it).
Let's try to apply this principle to the Pokémon franchise! Assuming you make the player pick a trainer class or something, you could - depending on the class they picked - offer a couple of different introductions to the story. And this is just one possibility, you could also add some kind of random factor, like in the Megaman Battle Network series (I only played the 4th one, but c'mon), even though this might put off some players. Split story lines and decision-based sidequests are other, more common means of adding a replay value. Unlocking higher difficulty-levels or other bonus-features might as well be interesting.
This one - also inspired by Dragon Age - is less practicable for Pokémon games, but I'll put it down anyways: Assuming the player only has a limited amount of... let's just call it "time" for now ...to get to know the important characters, they might want to know about the others, playing the game again, just to get that done. On my first run of DA:O, I played with the party of Alistair, Morrigan, Leliana, like anyone else would have done on the first run. Heck, I even killed Wynne before she joined the party, just not to piss Morrigan off, since I was trying to achieve a romance with her... anyway: I used Oghren a bit, because he's funny, but on my second run (female, human mage btw. ) I tried Oghren, Sten and Zevran. That also was pretty awesome.
Next on my agenda: A moving, hooking and all in all interesting story line. This also involves interesting characters; Even though DA's story wasn't all that creative, it was inscenated almost perfectly. The characters, most of them ready to meet death, had me shaking during the final dialogues, the boss fight got my heart beating like hell and the bad guy really made me hate him, want to kick his FKN ass and yes, rip his head off - something not many games (except for Bulletstorm, maybe) have achieved.
3rd: Getting rid of random encounters. I know, it's one of the most essential things people mentally connect with Pokémon, but let's face it: it pisses almost all the players off in one way or another. Some kind of randomness and reaccessibility (is that an actual word? o.O) to the battles might not be so bad, not allowing the player to "clear" an area, however, just makes me sick.
Finally: Huge worlds not worth exploring. If the player is not rewarded with a cool item, funny dialogue, or at least a beautiful view, they will just get upset and stop 'experiencing' the world you created for him. They will simply start 'skipping through it' on the fastest routes, avoid any unnecessary backtracking and so on. What good is a large world to explore if there isn't even a reason to do so? One way to handle this is to get rid of the Routes as they are in the Pokémon games by implementing a world map where the player can choose where to travel to. Depending on what road they travel on, you can still have some form of random encounter (those shouldn't be like REs in the Pokémon games, rather like a 'on the road'-map, with an event the player might do once, or something like that).
Well, I think that's pretty much all I can think of right now. To sum it up: I don't like to be annoyed by a game, I want to be entertained. The line is very fine, yet graduent.