Alchemy, Armoror, Speechcraft, Mercantile, Acrobatics, are a most of the non-combat based skills and they are either bland or only used in combat.
Here are some of my ideas:
Non-Combat skills
Fishing- Simple and easy. Loads of great games have it (LOZ series, Dark Cloud 2). I don't really need to explain it, but it could be a good way to make money. Attach bait, through line, and then real it in on a visual cue of the fish bite.
Cooking- Little more complicated, kinda like alchemy only less customizable (You can't really cook random bits of plant together and hope it is good). Say you caught a fish, its no good to you raw except for alchemy. So you fry in a nearby pan. It now restores health and fatigue and is a more potent ingredient.
Crafting- Even more complex. This is more of an enhanced version of Armoror, you gather ore of a type(Deadric, Glass, Iron, Ebony). Kinda like the Madness Ore/Amber thing from SI only the player makes it. You go to a smithy and smelt the ore or a weapon of that type. Then you get usable iron to which you bring to the anvil and craft a weapon. You should be able to choose from a variety of blade and hilt styles in order to make weopons appear more unique. However if your skill is low then when crafting you may fail, this failure can be thrown back into the smelting pot so waste not.
Music- Simple button pressing minigame. Kinda like LOZ for the N64. Taken from the "Music as a skill" thread, I think that a bard should be able to do what a bard is able to do. If you are successful in playing a song others nearby will have a higher disposition to you, you could lul them to sleep or get tips.
I want to see you are ideas, you can either improve upon mine, improve upon ones already in TES, or come up with your own. If you don't like the idea of non-combat based skills don't post.