Eurgh minigames suck. They suck sweaty, hairy ballsacks.
If its narrative based then it's got to be a genuinely compelling story, not some armadilloty generic thing where you can guess how it'l end halfway through playing. And not one where you have to pointlessly grind away at killing things or spend hours treking about featureless armadilloholes for some llama who just happens to need something that apparently only the random stranger that is you can get for them. One of the best games I've ever played was Planescape Torment, its from the 90s, it had fairly dull mechanics (outside of the dialogue system) and graphics were not great. But I was hooked in straight away, it had a pretty unusual start and since the character had some form of amnesia the story was a real mystery. And that mystery was kept up the whole way through, seriously it was amazeballs. I'd even go as far as to say totes amazeballs.
As for sandbox games I usually get bored after a while if there aren't many goals to achieve or only one way of achieving them. Take mount and blade, its gots some pretty awesome mechanics but after playing it for a while its gets boring pretty quickly. The only aim you can set is conquering calradia, and to do that you're going to have to fight a horrendous number of remarkably similar battles (and struggle through the sieges, which are by far the most frustrating and portugaled up part of the game.) I've never bothered to conquer calradia, hell I've only had one game that lasted long enough for a faction to be wiped out.
Actually it turns out two of the best games I've ever played come from the 90s, King of Dragon Pass is a very strange one, its a mix of sandbox strategy and narrative based rpg. Its entirely menu based but the mechanics are still quite interesting and it really does make you think. It has a sort of random event system that really keeps you on your toes and its so immersive that I could happily spend hours playing it through and completely ignoring the mainline story.