Lesee... how about the following:

A Magic Carpet mount. Bonus points if it includes a blue Genie who will turn you into a prince[ss]
A lion mount. No, not the warlion. How about a beautiful majestic animal that looks as proud and noble as a lion should?
A broomstick. Great for wizards and muggles alike.
Lalafell race exclusive mount: Roegadyn jogger. Ride on the shoulders of a big strong lad. Adorable, and convenient.
Miqote race exclusive mount: Ball of Yarn. What's not to like about rolling around on a giant ball of tantalizingly loose yarn?
Roegadyn race exclusive mount: A unicycle. Because it'd be hilarious. No other reason needed.

...I can't think of anything remotely amusing for the Hyur and Elezen races. They're a bit too vanilla to decide on.