This idea has been stuck in my head ever since I started playing again, though I'm not even sure if they would be possible.
My first idea was having a chat room like experience with your linkshell(s) on your phone. So you could talk to your friends and keep up with your social shell(s) even if you couldn't be near a pc or console.
My second mobile app idea was having an interactive Auction House app. You would be able to view the Auction House and make purchases from your main character's Gil supply and have the items sent to your Mog House. I don't think this one would be hard to do. I mean FFXIAH.com already does this without the ability to make purchases.
I just like these ideas as they would allow me, and you, a way to play my favorite mmo while on lunch at work or when I'm not at home.