Maybe we've got this backwards. Instead of looking at ideas for attributes and the things they can do, we can try looking at mechanics that are already in the games that are better governed by attributes than skills or item stats.
We've already got health, magicka, and stamina. And separate regen rates for all three. Carry weight. Stamina cost of sprinting. Stamina cost of power attacks. Stamina cost penalties from the weight of worn armor. Movement speed penalties from the weight of worn armor. Stamina regen penalties from running. Stamina/Magicka regen penalties from the weight of worn armor? Organize them all into an attribute set that makes sense and doesn't give a heavy imbalance to certain attributes. Don't look at it thinking "well, we have to put an intelligence/awareness/whatever attribute in there somehow, so how are we going to justify it?".
A weird, spur-of-the-moment idea that I haven't completely thought out: a weird hybrid of Skyrim's skill trees and Fallout 4's nonlinear perk chart. Every perk in the tree has an attribute requirement, and you can take them in any order so long as you have the right attributes. Additional ranks in that perk require a higher skill level, though (instead of a level requirement like Fallout). So a skill tree for a melee weapon type: INT requirements on the crafting branches of the tree, STR requirements on the power attack branches, AGI requirements on the attack speed or parry perks, you get the idea. Doesn't have to represent every level of every attribute in every skill tree, but it still requires a big rethink of how attributes are handled in Elder Scrolls. I'd also suggest going easy on criticizing attributes that have more purpose in gating perks than they do in their base effects, like what Fallout 4 does; that's still a big way to define what a character can or can't do, and it's not a bizarre thing in RPGs for attributes to be tied into other parts of the character system to give them more weight.
Stray thought about Strength requirements: They're dumb and arbitrary. But something I really liked about Fallout 4's system was that it managed to avoid STR requirements or penalties on equipment, but still managed to create the same end result by putting the Heavy Gunner and Steady Aim perks high up in the STR line.
Seditious thought: Speech shouldn't be a skill, at least considering how it was used in Skyrim. Humor me for a second. In Skyrim, every character bartered and every character picked on the speech checks regardless of whether or not they had a high speech skill. There's no reason not to, and without the class system (which should stay gone) there was really nothing to prohibit characters from developing their speech skill just by playing the game. They should still incorporate the perks somehow, particularly the ones that add new options like fencing to any merchant, investing in merchants, or the new Intimidation perk in Fallout 4. But hey, this would be a nice way to fit in a Personality attribute in the weirdo Fallout-Elder Scrolls hybrid I sketched out above. (incidentally, this same logic could be used for lockpicking, which could potentially fit into Agility or Dexterity or whatever >.> if it fits i picks)
I want to see them expand on mounts and mounted combat. Make it a little less frustrating to use them (damn horses constantly jumping into fights or moving so fast downhill they leave the ground and ragdoll you), put more NPCs on horses, and have those NPCs fight each other on horses! "You got Mount&Blade in my Elder Scrolls! Ew!" "You got Elder Scrolls in my Mount&Blade! Yum!"
A "Radiant Economy" would be pretty great, too. Complete those Mine Ore/Farm Crops/Chop Wood quests to increase the amount of vendor gold in a region. Complete radiant quests for merchants to increase their inventory, or even engage in a little unfriendly competition to improve the economy in one region and reduce it another. Turn mills, mines, and farms found across the province into quasi-Workshop settlements that we can sabotage or repair to make an impact on the economy. And maybe put in some systems where bandit amass more loot as the regional economy suffers.