A feature that Blizzard fans have asked for for a long time, is the ability to change your BattleTag. That's finally come about, so kudos for Blizzard to finally listening, although not everyone is happy about the implementation, as although Blizzard offers one name change for free, after that it's asking for $10 per name change
So if you do wish to change your name, think long and hard about what you want it do be. If you pick something silly again, you'll have to start paying to fix it.
Of course Blizzard will not allow any new names that breach its naming policy, but unlike a lot of services your BattleTag doesn't have to be unique. You're uniquely identified with some other metric behind the scenes, so you can call yourself whatever you like. There might be other people with the same name if you go for something common, but it's unlikely you'll bump into them too often.
A neat little feature of the changeover is that nothing in your account will change with it, so even if you change your name, you'll still retain your level, all your unlocks in various games and your friends list will remain intact.