Name Change should show your previous name to your friends in their Friend List
If you have a lot of people in your Friend List, you often don't notice when a friend has changed their name. You suddenly see a weird name in your friend list so you have to ask that person if he changed his name or if he's some random dude that added you after a game to chat with. This is something that should be unnecessary.
I think it's pretty straight forward that if you change your name, your previous name should be shown to everyone who was friends with you at the time of the name change. Simply have the text "Previously known as: ..." shown under their name for a certain period (1 or 2 weeks). Alternatively, you could instead add the option "Show previous name" when you right click the name in your Friend List. I prefer the first solution, though.
The picture below is just an example, created with my l33t Paint skillz.