The trick to a minigame, imo, is making it an interesting game without an obvious best strategy. The best I've come up with was the research minigame for BtVS (
link). While doing research is just a stat + Skill roll to get success levels, I expanded it into a larger system. (In CineUnisystem, you roll a d10 plus a skill plus an attribute. 9 is a basic success, with higher numbers equaling more successes. 9-10 is one, 11-12 is two, etc)
First, I defined success as discovering a Clue. This could be the weaknesses of a demon, the purpose of a ritual, or what have you. Clues had two properties, difficulty and obscurity. Difficulty reflected how much detailed knowledge one would need to find the answer. This was rated in the number of successes needed to find the answer. Obscurity was sort of like armor for the clue - for each point of obscurity the first success on a roll didn't count. So if a PC investigator rolled a 14 he earned 3 successes. But if the obscurity is 2 he only applies one to the total needed to find the answer.
The next part I stated that you can't simply have everyone roll Int + Occult and total it up. That's now how teamwork operates. There were a variety of options, and each PC that wanted to cooperate had to doa different one. They were named after events on the show for fun. So the main researcher would
Get My Books, Look Stuff Up, while a second could
Cross reference, adding his successes to the main roll. A third could
Consider the Quantum Mechanical, rolling Int+Science or Knowledge to add successes. Each round of rolling took 4 hours game time, so one character could
Keep it Organized to keep them from losing a success if they rest. And if you have no applicable skills, you can always
Go for Donuts, giving everyone a +1 on their rolls for the round.
The final piece was the effect of narrative on the process. The Obscurity was flexible, and would vary depending on exactly what they were researching. For example, let's say that a witch is casting spells to turn people into mad wolves. At first, the character might assume the problem is a werewolf on the loose. Researching werewolves, the GM assigns an obscurity of 5. This makes success near impossible, because the players are barking up the wrong tree. After some more attacks and clues, they find that the transforming wolves don't respond to the full moon, and there are signs of rituals being done in their homes. They switch to looking for a spell that does this, and the GM drops the obscurity to 1. Still a bit of an obscure spell, but they're on the right track now!
Of course, Obscurity is secret information. They don't know that until they start rolling.
One of my players remarked that the system was 'like fighting, only with books!'. I think that should be the goal for a good minigame.