Flat battles, the Pokéthelon, and the Following Feature happen to be the top three features I like the best. Flat battles like kohei stated were long overdue since Diamond and Pearl. It'll now allow so many to stick with their original team they designed and easily downgrade the levels to 50 for Nintendo Hosted Tournaments and/or for the Wii, without the hassle of retraining anything. So, definitely a plus for HGSS that I think many will appreciate right now.
The Pokéthelon because of the challenge it brings. It shares a variety of games for you and your Pokémon, plus with the inclusion of stats it really adds a bonus to the difficulty level of the feature. Even if they didn't include Pokémon Contests like I had hoped, they still brought in something new that looks promising enough.
As for the Following Feature, what can I say; it is just great for me all around. A return from Pokémon Yellow, but definitely enhanced. It includes all 493 Pokémon plus the shiny forms and alternate forms such as Shaymin and Shaymin's Sky Form. I don't think it could have gotten any better except for some improved OW sprites, but nothing is ever really perfect.