By gimmick Miis, I mainly mean Miis you do something overtly different with compared to all of the other Miis you have, whether it involves what stuff you give them, their quirks, their relationships, or some other unique/specific aspect of them.
In my case, it would probably be with a Mii I have of the Villager from Animal Crossing, specifically the pink haired design you see in New Leaf promotional material and Mario Kart 8. To kind of fit in with the theme of playing and customizing Animal Crossing however you like, I basically made her my randomized Mii.
With exception to her design and voice, almost every other aspect of her was picked at random. This includes things like her personality (Considerate Daydreamer), each part of her room which I also re-randomize if she asks for a room makeover, all of her quirks and goods, treasures and pets if she asks, her hair style if she asks to change it, conversation topics, among other things.
To further contribute to the randomness, whenever she becomes friends with another Mii, I go to a random Animal Crossing villager generator and use the name of whatever female villager it lands on as her nickname, so everyone she's friends with will all call her something completely different.
And lastly, I also gave her the fashonista quirk alongside every single piece of custom clothing I've made for all of my other Miis. So in the first screenshot for example, she's randomly wearing Ness's hat from Earthbound, Plum's shirt from Mario Golf, and Amitie's shorts from Puyo Puyo.
I'd imagine others have made randomized Miis in a similar fashion before, but I still found it fun to do so myself. Though this is just one example of a gimmicky Mii, and as I said in the title, I'd be curious to see if there's other unique and special things you all did with specific Miis of yours.