According to a recent post on battle.net, it's now pretty simple for StarCraft 2 players to change their character name, as well as free.
Blizzard promised players the ability some time ago, but seemed to have forgotten about it, setting out to work on Battle Tags and Heart of the Swarm in its place.
Here's the post in full. If you'd rather just skip ahead on how to change your name, it's the italicized bit at the bottom.
"We owe you an apology. Some time ago, we communicated a plan to make a paid name change service available to players. We want you to be able to change your name in-game more freely, and do so as often as you like.
The small percentage of players who used the free name changes that we previously offered, the introduction of Battle.net Battle Tags, careful consideration of how character names will work with Battle Tags in the future, and our increasing focus on completing StarCraft II: Heart of the Swarm all contributed to us deprioritizing the Paid Character Name Change feature. The fact that we didn't keep you updated when we deprioritized the feature was a mistake, and we aim to do a much better job of keeping you informed in the future.
We sincerely apologize for any inconvenience that the lack of name change options has caused anyone. We've definitely heard your passion for this subject, and we should have more information about a Paid Character Name Change feature shortly after the release of Heart of the Swarm. In the meantime, we've added a new free name change to all player accounts.
We're planning to offer another free name change to all players when the new Clans feature goes live in the next couple of months, so don't worry about removing your clan tag from your character name with this one. We will alert you to that when the time comes.
If you'd like to change your character name, log in to Account Management and choose your StarCraft II: Wings of Liberty game license. On your game license screen a "Character Name Change" button at the bottom will initiate the request. After confirming that you'd like to change your character name, log in to StarCraft II: Wings of Liberty, and you'll be prompted to assign your character a new name."
