Nope, it has no significance, whatsoever.
Leading up to the original release in 2010, long before the game was reworked for A Realm Reborn, they had talked about potentially tying your character's nameday into the game in some way, but nothing ever came about. In those pre-2.0 days the current Eorzean date was displayed in the UI, next to the weather, but this was left out from the 2013 re-release.
At this point it seems very unlikely that it will ever be anything more than a bit of auxiliary information on your character's profile. However, if at some point in the future they add something that uses your nameday, and you feel that you've chosen poorly, you're not completely without options. Using a Fantasia will allow you to remake everything about your character, except for their name.
You'll receive one free Fantasia after 30 days of paid subscription (the 30 days of free play that come with purchasing a copy of the game don't count). You'll also receive one as one of the bonus items in the Heavensward Collector's Edition should you choose to buy it. And finally, if you have used up your free ones and feel it's important enough, you can purchase Fantasia on the Mog Station cash shop for USD 10/GBP 5.75/EUR 7.