Rescuing all three Jigglypuff in the Cave track. For saving them, they'll perform for you nearby the exit. Even just rescuing one of them will get you adorable results, though of course it's best with all three.
Zapdos restoring electricity to the Tunnel might count, especially since the background music becomes less ominous.
Toss some food to those two Charmander in the Volcano and watch as they call over their four friends to share the spoils. Your reward is (if you can get them to move to the right positions) a shot of all six of them marching in unison to the Poké Flute.
Pikachu watching Diglett pop out of the ground is really cute, as the latter looks absolutely delighted every time he sees his friend and will ignore any Pokemon food thrown nearby in favor of continuing to play.
In the Cave, hatching Articuno will make it fly behind Todd near the end of the stage. There's nothing special about this, but there is a reason for it. If the player rescues Balloon Pikachu as well, the Pikachu will be riding on Articuno's back. The fact that Articuno was nice enough to give Pikachu a ride is nice on its own, but the fact that he was given the rare opportunity to ride a legendary of all things is truly heartwarming.