It's hard to imagine how it would work, though. The skill-boosting quests already in game are either gathering missions (fishing, mining) or cases where you prepare a number of recipes (potions, meals). How would repairing work?
1) Can't ask for repaired items of certain types, as the player could just hand over any items at all and never use the repair skill. Just buy or loot the required items and that's that.
2) If the NPC gives out items to be repaired it might work, although I'm not sure if the game can distinguish between repaired and unrepaired items.
3) The NPC could hand out special items to be repaired, but as above I don't know if the game can tell if something has been repaired or not. I suppose if the base item give is 1/50 the game could check to see if it's any better than that. Or check base value to see if it's increased. The time taken to repair these could be longer than normal, making it take real effort to complete. Maybe make them quest items?
That third option might work, but only if the game can check quality or value of the returned item, and it would still need to have much longer repair times than normal or risk being too easy.
Still, if that's dealt with it shouldn't be too hard. Start off repairing training gear, and as your skill improves you get to work on better equipment.