When I started playing league, I had some pain issues. With a few adjustments and a little bit of time, it stopped being a problem. Here's what I found helpful:
Gaming mouse. If you don't already have one, invest in a gaming mouse with a good ergonomic design. I spent about $50 for a gaming mouse and found it worth every penny. Compared to the $10 mouse I previously had, it was far more comfortable and a much better fit for my hand.
Arm positioning. Bad arm positioning puts unnecessary stress on your hand. If you're angling your elbow or any part of your arm up, that's bad news. You want to have your arm and elbow resting on your desk. Check out some videos of gaming pros and watch how they position their mouse arm. It's difficult to describe, but their mouse arm serves as an anchor for their hand. Not only is it more comfortable when you get used to it, but it can also help you move your mouse faster.
Game settings. I like playing marksmen, probably the most taxing type of class to play in terms of hand stress because of all the positioning and kiting. In my game settings, I bound the player attack move click command to my "R" key (my strongest keyboard finger) so that I can auto attack whatever is closest to my cursor. It saves me hundreds of clicks per game because I no longer click on minions and enemies for an aa--I just hover my mouse near them.