I think it would be best if you let the player decide their own name :/
If the tree-based Professor wants to protect the Pikachu from Team Rocket, surely he would give it to an experienced Trainer/ Gym Leader/ E4 Member/ Champion if it was so special rather than a newbie due to the fact that it holds the fate of the world in its hands. Any basic TR member could waltz up, pound a begining trainer into the ground and walk away with a shiny-new Pikachu to use in their devious plans.
I just can't see the tree-based Professor as being in a perfectly sane mindset when he made this decision.
This is where I suddenly realised you're copying directly from the Anime. While it isn't necessarily a bad thing, I just feel like there have been to many scenarios (Not speciffically in ROM Hacking, though) where a person is given a super-Pikachu and plays keep-away with Team Rocket. Branch out, you don't have to take issues directly from the Anime and translate them into a new media form.
Besides, since this Pikachu is super strong, it would kinda' ruin the beginning of the game since its so overpowered.
That's kinda' what happen when you distrupt the natural order :P
Wait, I thought that TR had stolen the 'Balance' Pokemon AKA, Rayquaza - Why is it suddenly on the top of a tower? Shouldn't it be in some sort of holding-facility or something, given thats how TR generally hold their Pokemon?
As I stated earlier, this idea seems to be borrowing more than just a page from the Anime's book, whilst the ending is emulating the conclusion of Pokemon Emerald. It feels like a great mish-mash of the two, with nothing that makes it your own. A greater knowledge of what you're planning to include in this idea would have been better for me to understand what's going on between you getting Pikachu and going meeting with Rayquaza at the top of the tower - Right now there is no mid section of the hack, which I would have liked to have see.