Been putting some thought into this for the past few days, and here's some of the things I would like to see:
Secret Bases: Already been talked about in the other thread, but this was a silly, fun little sidegame, and, in OR/AS, it would have the potential to make use of StreetPass content and maybe also have a changing room for people to change clothing (if trainer customization is brought back).
Contests: I really loved the Contests in Gen 3 (Gen 4's were a bit too awkward and fiddly); they gave you a reason to use Pokémon and moves you might not otherwise and were actually somewhat strategic, as opposed to many of the cosmetic sidegames in later gens (*cough*PRvideos*cough*). Similarly, we'd have to have berry-blending again, of course, and therefore berry plants everywhere *nodnod*.
Plot: As much as the different storylines in Ruby and Sapphire were cool, I'm hoping that OR/AS will follow the HGSS pattern of including some of the elements from its 3rd game. I think it would be best if both teams were shown in a villainous role and if Rayquaza plays a more integral role, perhaps having something to do with the Alpha/Omega forms of Kyogre and Groudon. I could expand on this, but it would get kind of fanfictiony. I think I would prefer Steven as the Champion, though, since it felt more fitting in the storyline.
Mechanics from Gen 6 That I Want: I'm on board with everyone that's said they want trainer customization back (preferably with the option for no hats >_>), but another new feature that I'd really love to be included in OR/AS is Wonder Trading. It's silly and you get mostly crap Pokémon, but I've been having so much fun with it, especially in my current "WonderLocke" playthrough. Super Training should also stick around, so that when I'm finally unlazy enough to make an actual competitive team, it won't be difficult.
Mechanics from Gen 6 That I Definitely Do Not Want: O-Powers can stay in XY, please. Not only did they feel metagamey and 4th wall break-y, but Mr. Bonding was creepy. No thank you.
Mechanics from Gen 6 That I Don't Really Care About Either Way: Pokémon Amie. It was cute, but I didn't like the extra powers (criticals and what have you), and the only thing it was really useful for was Sylveon. You couldn't evolve Leafeon and Glaceon in HGSS, so there's no precedent for having Amie in the new games just to be able to get Sylveon. If it's included, I probably won't bother, but at least I won't have a creepy guy in a purple suit waggling his eyebrows at me in every new town.
Pokémon Riding, Following, and Other Interactions: In HGSS, we got Pokémon following behind you. In X/Y, we got to ride Rhyhorns, Skiddos, and Mamoswine, and if we used Surf with a Lapras, you could see the Lapras in the water. I would really super love to have this expanded in ORAS. I definitely agree with the people who want your Pokémon to follow you around again, and I would also love if it we got more unique animations for using Pokémon with HM moves, such as Surf, Fly, and Dive.
Speaking of Dive... This is one of my favourite things about the original gen 3. I hope that Diving is expanded and played around with a lot, because there are so many cool things that could potentially be done down there.
Post Game: I would love some post game content. I guess BW spoiled me in this respect. Battle Frontier would be cool, I guess, but I'd prefer more of the BW or HGSS-style post game where there was more stuff to do.
Safaris: I loved the Friend Safaris of XY, but I think it would be a lot of fun to have the regular Gen 3 Safari Zone again. Perhaps we could have both the SZ and the FS, or perhaps do something else with FC connection content.
Legendary Availability worked into the storyline: I loved that there were tonnes of legendaries you could catch in BW2, but it felt kind of weird to me that they just appeared without being hardly mentioned or talked about or anything. I think it would be really cool if there were sidequests (some of that delicious post-game content I talked about maybe?) that actually talked about the various legendaries you could catch and made something of a story/quest out of it, instead of just battle, catch, done.
Aaaanyway, I think that's it, for now. >.>