My favorite is when I write down all the skills and races on tiny pieces of paper, choose a race at random, and 6 accompanying skills. For instance:
Nord - Smithing, Destruction, Illusion, Block, Lockpicking, Alchemy. No armor aside from a shield, only damage dealer is Destruction. Can use Alchemy for health potions to help stay alive, and Destruction damage buffing. Will probably rely heavily on Illusion to stay alive, but can't use sneak.
Argonian - Enchanting, One handed, Conjuration, Restoration, Light Armor, Speech. Looks like a standard Death Knight/Paladin build, can either Dual Wield or Paladin style sword/Resto. No Smithing, so getting good at Enchanting and Conjuration will be important for melee damage soaking.
High Elf - Two handed, Sneak, Heavy armor, Restoration, Lockpicking, Light armor. No crafting skills on this one. You could go a Resto warrior, or a sneaky guy with giant hammers, or some combination of that.
I don't really like when I get one like Race - Pickpocket, Speech, Restoration, Light armor, Heavy armor, Lockpicking. I guess that could be your pacifist or unarmed guy.
I haven't played these builds, I just drew them right now. But I get a lot of great and interesting combinations that keeps the game very fresh. You could even limit yourself to 5 or 4 skills, or choose 4 of the 6 you drew to use.