I know a lot of people have been complaining about this ever since release, so this is my idea on how it could have been done:

The class could develop around your character (to associate with the new development/leveling system). That is, the more you play a certain group of skills, the more aligned you become towards a certain class/name. This could be feasible as many of the stats that you have (favorite shout, favorite weapon, etc.) are already measured.

Over time...

One Handed + Destruction: spellsword
Two Handed + Heavy Armor: warrior

The perks you would use could factor in as well, seeing as necromancers would want to be called as such!

Destruction (flame) + Conjuration (raise dead): Pyromancer (?)

The only downside I can think of could be ill-named ambiguous (if you will) classes (so much so that you may feel obliged to orient yourself in a specific manner.) and measuring perk use... but I still think it's a favorable idea.
The thing is, people already do this on the forums, so maybe a logical implementation in the game could stifle some of the seemingly perpetual groaning.
You may ask: why didn't Bethesda implement their old system immediately? Well, and this may not need to be said, Bethesda is moving away from what you would call an "UGH" moment for the lack of a better word/phrase (feel free to correct me on this). They don't want people to be at level 59 and thinking: I don't like being a destructionist, I want to be a warrior, oh but I'm named a "mage".I suppose I'll just have start over... I don't know about you, but this happened to me in Oblivion, so the ambivalence is a bit of a relief in Skyim, even if it does feel like a bit of a foreign idea in an RPG...
The core purpose of this idea is so the "UGH" moment won't happen, and you can be a level 30 "Brute" at one moment, and a level 73 "Archer"

So, in conclusion:

• Developing-with-your-Character Class System
  
• Additional Perk-Alignment
  
• Possible Unwanted Naming and/or Logic Problems as well as many possible name differentiation problems. (maybe you want to be called an Archermancer or such, seeing as just "Archer" would be more favorable for the sake of memory and logic.)
  
• Maybe this is an unnecessary bullet, but... Changing Classes Mid-Game

There's also simply naming the class yourself in-game, but who ever heard of something as simple as that?!

If this idea has already been posted (I felt around in the forums only so much), feel free to comment as necessary.

I actually wouldn't mind the game changing the name of your class as you play. If I remember correctly there is a game that does this really well, but I can't seem to remember the name of it.

What is the point? If you want to be a warrior, or a spell sword, then call yourself that. Why do you need an arbitrary system to tell you what you are? The point of the game is to play how you not, not how a label defines you.

Try telling that to Merari.

Mainly for role-playing purposes. You'd be surprised how popular that is in a Role Playing Game.