I did this already with the Elves and the Halflings (Elves have only a limited time in the Realm of the mortals and Halflings just try to earn enough money and experience to become entrepreneurs of sorts). With the Dwarves I had a difficult time coming up with something concise that wasn't to far of from what would be recognized as "dwarven"*. This is my take on it.
|Cute, right?** (source)|
"I'm Dürggal from the Mountain Karrakas!"
Easy enough explained: Until they reach their level cap, dwarves aren't considered clan members. So only with level 12 they get fully recognized by the clan and their full name! Until then, when introducing themselves, they only give their first name and the mountain they are coming from. This is their reason to go on adventure, they want to be full members of their society.
There's even a possibility for lost and banned dwarves, who give themselves "wrong" names (could even be an adventure seed...).
*One idea being that a Dwarves beard was a symbiosis with a fungus of sorts. That's why they couldn't shave. But anyway, that's for another day. Maybe...
**Haven't made up an opinion about this picture yet. Stumbled across it some time ago and thought, well, there's something to it. But a hedgehog? Really? And what's with the lamp hanging on the backpack? Anyway, here it is.