Account sharing and phishing incidents make up the bulk of account recovery cases.
I know that its tempting for players to share an account with bro, sis, bestie, girlfriend, boyfriend, husband, wife, etc etc. But what happens when one of these relationships have a falling-out? Now someone has access to your juicy game account and the person doesn't like you much. Not a good combo. :(
For phishing, many players get tempted by free RP, or other social engineering scams. The *only *place you should be entering your LoL username and password are within the client and on leagueoflegends.com.
The easiest way to better secure your account is to verify your email. This can be done when logged into leagueoflegends.com, and you get a free 4-win IP boost by doing it! Also, you should have a different password for your LoL and email account.
As for the best way to prove ownership, I really can't say. First because every case is unique, and second because the bad guys can read this and adjust their bad ways accordingly. If you're dealing with an account recovery case, we will ask you privately for all the information we might need to resolve it. :)