I personally did work on modeling injury risk for an NBA team, which at the time was something they hadn't done much with and found very useful. Unfortunately it might be hard to apply that to LoL.
> Obviously the games are different, but perhaps there are some common themes or generalized structures that can help frame what questions to ask?
One of the most important things to understand is what the front office, coaches, and players want to know. Like I said, much of analytics these days (and not just in sports) is a solution searching for a problem. We build models based on what we can with the data, but the output of the models isn't actionable or something anyone is asking for. A better approach is to start from the questions that loom big in the eyes of those you are doing the analytics for. That sounds simple, but often doesn't happen, either because communication is poor, or neither party really understands what the other is trying to do. Also, often times a team doesn't even really know what questions they'd like answered, or at least not enough to verbalize it well.
One thing I've found consistently useful is focusing on measuring uncertainty. These days people are extremely excited about machine learning and "AI", and I work extensively with deep learning in my current position. But most times in analytics, especially sports analytics, simply getting the best prediction possible isn't actually useful. What is far more useful is measures of uncertainty through more traditional statistical methods, especially Bayesian statistics and Monte Carlo simulation. It's one thing to predict a player in the draft will be better than another. But a team usually knows the answer to that already anyways. Its far more useful to quantify the range of possible outcomes for a player. Is it practically a sure thing? Is it not a sure thing but a reasonable bet? Or is it really a toss-up? These are insights that teams often don't have but can find much more useful in their decision-making.
I generally don't focus on in-game strategy, since it is harder to give the team something they don't already know that they can reasonably act on. But meta-game insights could have a lot of potential. For traditional sports, this can be things like assessing player value, injury risk, draft capital, and contract valuation. Some of these may not translate well to LoL. I haven't played the game, but I've watched it a few times, and perhaps something could be done looking for insights regarding the hero pool. There are so many different heroes in the game that can act/oppose each other that quantifying those relationships somehow could be interesting.