Well, the first thing I would change would be the disappointing final "boss" battle (and I hesitate to even call Red a "boss" as he is pretty pathetic as far as trainers go). I'm not gonna go as far to say that Red completely ruined my Soul Silver experience, but he certainly did bring my level of enjoyment down a few notches. The issue with Red I have is twofold: First, he doesn't have his Pokemon all fully evolved (and the one he doesn't just happens to be among the weakest and stupidest-looking of all Pokemon known to man) which is a big no-no; it left a huge opening in his defenses and made him absurdly simple to annihilate. Second is that he kept my favorite trainer (who should have been given her shot at appearing in the game) a complete no-show. There was absolutely no excuse for Leaf's absence aside from the developers either being lazy or playing favorites.
My preferred solution would be to simply replace Red with Leaf and give her a team of diverse (read, no overlapping types) Pokemon all in their final evolutionary stages. Unlike Red, she wouldn't be a complete pushover. Barring that, the DS dual slot could have been used to determine who you face in Mount Silver and what team they'll have, depending your character and party in your GBA game (with a default boss in case you can't use the dual slot). At the very least, the final boss should have been someone who has the intelligence to fully evolve their Pokemon so they pose some semblance of a challenge. As he stands, Red embarrasses and brings shame upon the otherwise solid game called Pokemon HG/SS.
Well, with the worst part of HG/SS out of the way, let's move on to other niggling problems.
*Rock Climb was a stupid addition. Did it really need to be included?
*The Safari Zone. No matter how much you try to sugarcoat it, the basic Rock/Bait formula for Pokemon capturing simply doesn't work. I preferred not having a Safari Zone in the originals.
*Please, let us delete Pokegear phone numbers. If you accidentially get Joey's number there's no way to rid yourself of it and you'll be regretting it for life.
*You should be able to select one Pokemon in your party to follow you around regardless of whether it's first on your list. With me changing the order of my Pokemon constantly, that feature lost a bit of its appeal as a different Pokemon would be there after every battle.
*Dear Mom: Stop embezzling my money to buy worthless berries and then pestering me over the phone whenever you do. That system worked a lot better in the originals because her purchases weren't nearly as frequent back then. Need I remind her that embezzlement is a felony?
*I want to acquire real estate and decorate it lavishly, like the Villa in Platinum. The beach west of Olivine would make for a nice setting.
*The Celebi event battle with Giovanni? Yeah, he should have been tougher.
*Did the female protaginist need to look like Mario with a balloon hat? In fact, why can't we have customized characters? And yes, I'm going to ask that of every Pokemon game from here on out.