Pokemon GO is blowing up worldwide in the areas that it has been released in, and even outside of those areas through people sideloading the APK. To say that the app's popularity is based on its features would be almost entirely accurate, since most players want to find and catch Pokemon against the backdrop of everyday locations, but there are a good number of features that could do some good if they made it into the game. Some have been promised by Niantic and some have not, but all of these would be fairly simple to implement into the game in its current form.
The first and probably easiest feature would be shiny Pokemon. Super-rare, differently colored Pokemon that appear at random with a very low encounter rate in the games would fit perfectly into the series' first mobile outing. The next easiest thing to implement would probably be holding items. From the second generation onward, Pokemon were able to hold certain items, which would induce certain effects when the Pokemon was traded, or would be used automatically during battle for things like healing, or buffing under certain conditions. Trading, which Niantic has already promised in a future update, may be a bit harder to put in. One on one battles with people nearby would be about as difficult to put in as trading, and is about as in demand as trading. Mega evolutions, temporary Pokemon upgrades found in the newest games on the 3DS would be nice to have as well.
Starting out the list of slightly more difficult features to put in, many fans would likely be glad to see Pokemon contests and everything they bring. From putting your Pokemon's singing voice on the spot to dressing them in the latest and cutest fashions for the judges, contests would make a nice distraction from the fighting and catching that the app currently offers. Growing your own berries would also be a nice touch, as would an official way to contact nearby trainers, even if only to show them your trainer card and current location, or to send quick predetermined messages. There are unofficial solutions for nearby chatting, of course, but an official implementation would be safer and easier. Finally, the crown jewel that would deepen the battle system significantly, some fans would love to see Pokemon in the mobile game have personality traits and physical properties like strength and speed that increase upon level up.