I would like having clothing be on an individual basis, not set outfits. The Tailor was almost useless in Red Dead Redemption.
I'm 100% in favor of an eating/drinking/sleeping cycle in the game.
A more oppressive inventory system.

It was kind of ridiculous how you could carry 22 weapons and an entire general store of goods on your back. I think having a limit of one weapon per holster, and a much smaller carrying limit, would be good. Of course, you could store more on your horse, a LOT more in a wagon, and could buy pouches and bags for more personal carrying space.
We should be able to buy wagons and save them at our safehouses. As a good player, I was disappointed at how rare wagons and carts were... you either had to get them through Random Encounters or steal them.
Followers would be a great addition, and could be implemented like the Friends system in GTA: you have to build up a relationship with them (through drinkin', whorin', gamblin', and huntin') and then they will join you as a companion. Dogs would also be good, mostly for tracking animals (instead of having to use the bait). They could even be directed to sniff out specific animals.
More jobs and activities! If any job or activity ever appears in the missions, it should be doable in the sandbox. Bonnie and Abigail made us do a million cattle ranching jobs, but it's not a side activity! And I would have loved to be able to horse and cart race in Rathskeller Fork outside of the West Dickens missions. Other things could be added; some people have suggested Old West shows, which I think is a good idea.
More crimes. Train robberies were non-existent, and we were limited to one bank robbery (which had to be a nighttime raid). There should be train robberies, bank robberies, rustling, exhuming, moonshining, and anything else you can think of short of rape.
Gang-building! This goes in hand with companions, but some randomly-generated NPC characters you hire. It would be an actual gang if you are low Honor, a vigilante group or posse if you are high Honor.
Bounty hunting was pretty good as it was, but it could be even more epic. What if the bounty hunt involved actual detective skills, where you track your bounty based on clues? The current system could be replaced with assassinations (bad bounties). Having the Pinkerton Agency feature as a job-giver would be great too.
If a farm or something similar does feature again, it should have jobs associated with it, or even actual managing. I understand why many of you don't want "Farmville", but if it's there, there out to be a point to it (and the player wouldn't even have to work it themselves, they could higher laborers). The problem is this: in the Beecher's Hope series of missions, the only farming we saw John do was in missions. Otherwise, it was all hunting. When I played through one time, I role-played and try to have John be a family man during the last stretch of the game. But it didn't make sense how we spent all his day hunting when there was cattle driving to do. I would pay for a Farming DLC that lets us actually make use of Beecher's Hope, and if the protagonist does own property in the next game, then I hope there will be special activities for it.
I saw that in Call of Juarez, you can use a lasso to reach higher grounds. Since the lasso is currently a single-purpose tool (with a very narrow purpose), I would love to have it be used for climbing.
Music minigames would be neat. It would be nice if we could go into the wilderness, set up a campsite, and then play on a harmonica/guitar/fiddle, or sit down at a saloon piano (job?).
Melee is the one biggest disappointment in the game. Something similar to Assassin's Creed melee would be great, and it would be nice if we had, y'know, more than one/two melee weapons. Besides knives, there could be a cutlass, a dao (Chinese sword, presumably brought over by immigrants), a rapier (like Zorro), etcetera. And we could actually have good bar brawls: an unarmed system where you can handle any item. Bar brawls would be nice as Random Encounters or activities: suddenly a brawl randomly breaks out, like in the movies, or you can instigate one and try to survive in exchange for a Fame boost (and maybe other bonuses).
Stealth is the second-biggest disappointment, since John begins shouting out trash talk pretty much any time you try to stealthily kill enemies. A stealth system, crawling on the ground, and a bow and arrow (much suggested) would all be fine additions.