It is about having a level playing field to the greatest extent possible.
That is such an impossible thing to achieve in a worldwide game like FFXIV. When the server launched online for the very first time, you might think you'd be there at the very second it goes online to reserve your name. But the gods of Square-Enix have different plan for you. A minute before the server goes online, your stomach complained about the chilli you ate the night before and BAM! You gotta go to the toilet. You tried to bring your laptop with you to reserve the super special name, but no such luck. Battery died on you, and your sister just accidentally let the dog ate your laptop's cable. You cursed and tried to find a bucket to sit on in front of the PC. But alas, the last bucket was already used to house the cats you just picked from the streets a few days ago. And your stomach was still complaining. So you had to go to the toilet. You tried to do the business super ultra quick. Crap! Ran out of toilet paper. But you had luck this time as you found yesterday's newspaper. You're finally ready to create your character, but then the school bus arrived and your mom yelled at you to not miss it or else!!
Upon returning from school, you found out that Bel Crono has been taken in every single server. Luckily, you were smart enough to improvise and go with Bel Chrono.
But seriously though, my point is that because the launch of the server happens at the same time worldwide, there would be people on one side of the world where they wouldn't be playing FF XIV at the very moment it starts because of real-life commitment (e.g.: work, school, going to the toilet, etc). So it won't be level playing field for everyone. Someone somewhere would be at disadvantage. If you (and everyone else) can understand and tolerate that because of such thing those disadvantaged players would miss out on hours of gameplay at launch, I think you should be able to understand and tolerate that beta players keeping their name is such an irrelevant matter.
I can understand that some people really love their names, and their character names would have special meaning. But seriously, there's always another name for your character that you can use. Think of it as a fresh start for fresh new character, rather than always reusing the same name for all different characters you play.