TL:DR - Apologies for the long read, but hey I cannot explain it in a shorter way, so read it or don't

What they should do is get rid of all the classes and instead, like in the single player games, have skills trees for each weapon and armour type and release spell crafting.
Your weapon and armour skills level up through use, and you use sky shards to unlock different abilities/passives associated with the weapon/armour/spell line.
Levels should no longer exist in the traditional sense, your levels are your weapons and armour skill levels, another words how many Skyshards I have invested in each of the skill lines. But I couldn't go and collect all the Skyshards at level 5 and then just put all of them in my skill slots. You first need to level up your weapons and armour of choice, each time they level up you get to apply a Skyshard to them.
At set intervals maybe every 5 or 10 Skyshards invested in a set line you get to branch off into specific styles;
E.G I am wearing heavy armour, I have invested 5 Skyshards in improving my use with heavy armour, now I get the option to divert into health, stamina or magika bonuses. These lines will take you into different skill sets.
However certain armour types could never do certain things, e.g. stealth is not available to heavy armour users, and mobility would be slower and use more stamina. But light would never be as good at soaking up damage as heavy but would give you the most mobility and the ability to focus or concentration (magika) more. Medium would give you decent survival and decent mobility, which in turn gives you more confidence when in melee (stamina), because you have the advantage of both mobility and survival.
Each armour line has more powerful attributes the further you go into it's line, but there is no reason to stop you having a bit of everything.
It would be similar for weapons, pickup a weapon and start training in it by using it. However you would learn new attack techniques and depending on the weapon type they would use a different pool of resources.
Melee weapons and Bows and crossbows (please introduce...how longs it been now?) would use stamina, where as magical staves need magika.
Staves would change in that all staves simply allow you to channel whatever spell line you have focused on, get rid of resto staves, destro staves etc.. All staves are the same, bits of wood that channel your inner magika, some do it more effectively than others however.
The more you use one type of weapon the faster or more powerful you become with that type of weapon. Heavier weapons will always be slower than lighter weapons, and you'd better train in a second melee weapon if you want to fight up close because, staves or bows would be useless in melee combat.
At certain intersections of the skill trees when you branch off, you are picking an offensive (weapons) and defensive (armour) style. These intersections have names associated with the path you take.
So imagine I am donning Heavy armour and wielding a Staff, I branch off into health and armour on my armour and so I am tagged as a knight. Now with my Staff I decide I want to focus on restoration spells, so I branch into restoration and am tagged as Healer. With these two tags my class name now becomes Holy Knight or something.
After investing more points in restoration I have been picking more offensive spells over healing spells, around this time I also start training in 2H because I need survival in melee fighting.
My 2H skill line now has some unique skills that are a direct result of my dedicated training in the restoration skill line. I now suddenly don't want to use a staff as my main weapon, I want to focus primarily on 2H, these new unique skills are awesome. My dedication to learning restoration spell crafting has allowed me to channel some of my power through my new weapon. I have lost my powerful healing abilities, but I have gained some great offensive spells and my class has now change to reflect this, I am now considered a Crusader or something to that effect.
Now these are just examples, but this is how they should have done classes and skills etc... in this game.
Also, seeing as levels would be tied to weapons and armour training, anytime you recovered Skyshards (respec) or whatever, you'd be starting a new skill line all over again, effectively being level one again, meaning you'd better go practice on something weak else you'll get your behind whooped.
What this means is; firstly it opens up a whole new level of replayabilty because one character cannot learn all skills, you can try but you'll always have a lot of mid-ranged skills. It's perfectly fine to be a jack-of-all trades, but you'll never be as focused as a dedicated trainer of one skill line.
The fact that you have unique skills opening up based upon previous training means the 2nd, 3rd 4th time you play you'll always be discovering new skills and ways of playing.
More to the fact with enough skills and paths to follow, it'd be rare to find two characters of the same build. Mixing and matching different armour, weapons and spells could give endless possibilities to play styles.
The game as it is now is pick a class and pick some weapon skills, then pick a morph or two. No need to start a new character because I can unlearn everything and train again.
This should also be how crafting works, focus on one or maybe two lines to get the very best training, focus on everything and be an ok jack-of-all-trades type.
I know they will never do this, but heck, what a game it would have been if they did.
EDIT: Also you would no loner put points into Magika, Health or Stamina at every level, these are directly leveled in relation to your armour and weapon skill choices.