Today, Microsoft has unveiled the ability to utilize the https://account.live.comweb site to better manage Aliases. Before today's change, the process was a bit convoluted and you could only add new aliases or rename old ones.
Even though you can effectively sign in with any alias you have defined in your Microsoft account, the new capability allows you to force an alias to be used as the primary. What this does is change the alias that is displayed on devices that operate using a Microsoft account (Windows Phone, Windows 8 tablet, Xbox, et al).
The rename alias feature has been eliminated completely, and instead you'll need to simply remove an alias and add a new one.