In the ARAM queue that arrived with the Freljord patch you can re-roll a champion you've been given by using a re-roll credit. How fast do you accrue these re-rolls? Is it based on number of games played or number of champions owned?
You gain Reroll points based on games played and champions owned. Once you've saved up 250 Reroll points, it grants you 1 Reroll, which is used to perform a Reroll in ARAM. You can only have 2 Rerolls saved up; When you have 2 Rerolls, you no cannot accrue Reroll points.
The formula for Reroll points gained is
Standard 30 Points
Bonus From Champions Owned
Note that I own 58 champions.
From the Patch Notes:
The only mention of the formula that I've come across is that a finished game nets you 30+Champions points, from here but I'm not sure how reliable it is.