The way I see it, anime has three main elements: anime production (direction, artwork, animation, music), writing, and story concept. I guess the latter two would be the "execution" that's discussed in this thread.
Of these three, it's easiest to separate good from poor production. While there can be some disagreement due to different tastes, it's generally straightforward to see that a show has a good budget and well-crafted visuals and music. Good writing is more nebulous, and is sometimes hard to make out if the production isn't very good. It does happen from time to time, and the results can be quite pleasing. These elements
While bad story concepts are easy to identify, I have no real idea how to tell what a good story concept is. While you can have a novel concept, that doesn't necessarily mean that it's going to be any good. Likewise, a very derivative story concept can still be used to make a pretty good show. Then again, I'm not sure what this has to do with originality to begin with. Can someone articulate what makes for a good premise?
Personally, while there's story concepts that can turn me towards or away from a show, it has little to do with how good the show is. For example,
Girls und Panzer is a totally derivative show with only one novel concept: tanks. I happen to love tanks and tank trivia, so I'm enjoying quite a bit. It's not a very good show despite decent production values because the writing isn't all that good.
Sengoku Collection has one of the most insipid premises ever - a game where female versions of Sengoku daimyo and samurai fight for secret treasures. Yet it still managed to get creative writing and production, churning out some really good episodes in the process.
Zetsuen no Tempest has a rather bad premise and the writing is pretty awful, but it's still entertaining despite that because the action and choreography is good.
In short, I don't think that the premise of an anime is all that important. The most it can do is to turn me off a show, but that generally has more to do with being uninterested in the way the premise is tackled than anything else.
For the OP,
Azumanga Daioh is one of the most prosaic premises imaginable:
Girls have wacky semi-adventures in high school. Highlights include talking about hiccups, having sleepovers and playing in a park.
It's still a damned good show because of its other qualities.