As you could guess, I strolled around here in search for wisdom, but all I found was people with problems and no solutions. Seems like some players had their mouse buttons inverted after some patching, and seems like my friend was made victim too. After some brainstorming, here's what me and my friend (nickname "Eduhell", so you pay your due respects someday maybe =p ) did:
We searched for a file with config inputs that would be changeable via "Notepad", and we found 2 likely files at the folder [...]\League of Legends\Config, namely "game.cfg" and "input.ini". Funny thing is "input.ini" won't show to me, no matter what hidden file options I enable, just to Eduhell, so maybe you need to create this file in order to change it! Or maybe if you don't have it then you don't have troubles at all.
Now here's what's really important: open (or create a new file) a the file named "input.ini" and change the line that has "[GameEvents]evtPlayerMoveClick=[Button X]" where "X" is the number of the button must be "1", if you want your character to move when you click the right button. If you try to bind a different key ingame you'll notice that "MB1" is the right button.
It should kinda look like this:
That worked for him, hope this is helps!
P.S.: You might think that if you just get rid of the file and the buttons will come to your desired settings... think again and DON'T delete the file, it didn't work. Just change the button to "Button 1".