Herbalism mini games. The world desperately needs more really detailed, realistic potion-making and apothecary mini games. Or at least I desperately do.
In all seriousness, the mini games I personally enjoy the most are point and click (for mystery games, especially), crafting of any sort (has a wide range of applications, but I had better be able to consume or otherwise use the items I craft in a meaningful way or else I will be one grumpy player), generally anything with a focus on knowledge-gathering or exploration, and the old standby stat raisers.
I agree strongly with the points Evy brought up about how connected to the genre as well as the story, as Shinoki mentioned, a mini-game is. In a story-focused game like a VN, the game sort of needs to serve the story. If the VN's is also strongly a stat-raiser or an RPG, for example, then those mini-games really stop being mini-games and instead become a part of the story. If that's the case then it's great...as long as that gameplay is done well. With tiny mini-games that don't effect the plot at all and can be skipped with little inconvenience (for example, the Tic-Tac-Smack mini-game in XOXO: Droplets) then there's not so much pressure on for them to be good as with a game that's a part of the plot. I guess it depends on whether you are using mini-games as a way to further the storytelling or just as a fun way to break up the reading, or something in between. Regardless, I think it's definitely important to make sure the mini-game feels natural for the setting and story. Adding something a little unique to make the mini-game your own is nice too, provided it comes naturally.
Something else I want to bring up that isn't brought up much with mini-games is...accessibility. Provided of course that reading and language is not a problem, VNs tend to be a pretty accessible format to people with different disabilities, at least in my limited experience. Depending on what it is, adding a mini-game can potentially limit this accessibility. A good example is any mini-game that involves clicking something within a time limit: these are really common in a lot of different genres because they are versatile and fairly simple in design, and a lot of players seem to like them. However, if they aren't skip-able, timed mini games can be really bad for anyone with poor hand-eye coordination or difficulty with fine motor skills, possibly making a game that would otherwise be fine and enjoyable completely off-limits to them. So that's another, less commonly considered reason to make non-integral mini-games something that can be skipped.
It's interesting, if perhaps a bit unproductive, to think of instances where unusual mini-game/genre combinations would work. For example, I associate mystery games with point-and-click mini-games most strongly, but could a dress-up game also feasibly be worked into a mystery game? Disguises are a classic aspect of the genre, after all. I can't think of a way a dress-up game could seamlessly be worked into a horror game (unfortunately) but could a shooter mini-game be worked into an otome somehow? Knocking out fan girls with tranquilizer for the too-attractive-for-their-own-good otome heroes? At the risk of derailing this thread, I feel like making ill-fitting mini-games and genres work should be some sort of game jam challenge some day...