This is a mechanic I've been toying with for Song of Ethera. I'd like to encourage puzzle-solving, creative uses of abilities, stuff like that. I fondly remember a session in AD&D 2nd ed where we were lost and it was night-time. We were all looking over our non-weapon proficiencies and spells, trying to find something to help us. I noticed I had "Astrology" as a NWP, so I asked the if I could use Astrology to find the North star, and therefore North. He agreed, and we managed to get out of there. That's the kind of play I want to encourage.

Play is divided into Conflicts, each of which has an initial rating of 120. (number is arbitrary, but I picked it because it has a lot of factors) The players have to make a certain number of successful skill rolls to reduce the rating to 0 in order to resolve the Conflict. Rolls do different amounts of "damage" depending on what attribute was used for them, e.g. a gate might take 40 damage from Dexterity rolls but only 24 damage from Strength rolls. The amounts of damage taken from different rolls vary from Conflict to Conflict.
The GM shouldn't sit idly by and throw Conflicts at the players, though. If a player comes up with something the GM hadn't thought of, and it succeeds, the success does double damage to the Conflict.

What stops the players from just doing the same thing over and over to rack up the required number of successes? The GM. It's the GM's duty to keep the players on their toes; if the same strategy keeps working, the GM's not doing his job.

Does this mechanic encourage the kind of play that I want? What are the worst things that could go wrong with it? Does the GM have enough to do, and is there a better way to keep the GM engaged in the puzzles?

And as a side note, after thinking of this mechanic, I realized that it's kinda similar to the way Jobs and Franchise Dice work in InSpectres. So, uh, thanks, Jared. :P

I wouldn't call it 'damage' as much as I would call it 'Success Points' or some other such creature. The damage strikes too much of a physical/combative connotation. If you want the players to puzzle more than fight, the terminology should reflect that I think, otherwise you might muddle some players. Nothing major, but it will stick in the back of their mind "Damage? I thought this was a puzzle solving game."

As you word it, you could use this system for bribing guards at the city gate, for example. Prehaps these guards are known for their dedication to their duties; thus, out right bribing them or other such attempts might yield a point or two ("Step aside for twenty gold? Lets see the gold." *PC hands over gold.* "Hey, thanks, its charitable citizens like you who make the world a better place. Now get lost before I turn you in, punk." *Stays in place*)
SO, to effectively get past the guard, you may need to learn some secrets, stage an elaborate plot, etc. Say the guard has a family member who is really sick- the PCs could work up a scheme that would naturally have several steps and leads up to a messenger delivering a notice that their loved one is about to kick the bucket. That could be enough to send the guard running, leaving maybe one other guard left the PCs know they can deal with on their own.
National Treasure is a good example of that, how Ben Gates manages to steal the Declaration.