1. Add gyms. Mostly because this isn't Gen 1 anymore and because there's the space and narrative for it. If not this, then Johto prequel.
Mainly just four extra gyms. Make these different types based on version just to make them unique experiences. This would most likely be either post-game to make the Sevii Islands worth it, or during to actually make the games feel livelier and less like a bland port. Fighting is that dojo and Ghost in Lavender.
FR would have Fighting, Ice and Steel
LG would have Rock, Ghost and Dark
2. Change Koga to a Dark/Poison gym. Ninjas are known for their trickery and underhanded methods. Why would Koga or Janine be above such a type anyways, especially when the Pokemon used by Rocket are either Normal or bugs?
3. Allow evolution of post gen 1 during mid-game as an expanded Pokedex as an achievement.
4. Use Adventures and the classic series as inspiration to revamp the story. Kanto is the blandest region because it's very much a prototype, even in the remake. Make Gary an integral part of the plot, use Blue from the manga as a secondary rival, etc. I believe the tournament-esque version of the Pokemon League championship in the original Kanto and Johto series is far more interesting than what the games did. Just have the Elite Four as the final contestants aside from the previous winner which would greatly enhance the game.
There. Another plot thread to have Team Rocket try to foil.
5. Reference Johto. The Gen 1 games get off because they're the first. Not the remakes. There's enough hindsight that allows for the new game to have it be referenced and alluded to, if not a part of the story itself. While a Gen II-esque exploration of Johto might not work, it could because we could see the gym leaders before Falkner and Whitney, if they're the same type, etc. The legendaries were in FRLG anyways, so it's not like there was a care for continuity anyways.
6. Make the Sevii Islands either a superior post-game or integral to the story. Give them gyms, make them a part of the overall Rocket arc. Do something neat that makes them fun to work with. Make them connect to Johto somehow and thus, lead into that league as a prequel to HGSS.