The name "Temujin" isn't particularly significant, and simply means "of iron," meant to symbolize her unyielding will and resilience. However, this has two hidden meanings behind it: Temujin is the birth name of Genghis Khan, and is also a historically male name. So it's meant to convey strength as well as intimidation, and clue knowledgeable people into the fact that the player behind Temujin is a girl. (I'd have tried to find something else, but repositories of 16th-18th century Mongolian names with their meanings are kind of in short supply, and I don't really know Mongolian, heheh.)
(Only one person has made half of the connection. Despite wearing conservative clothing and a short haircut out of practicality, I've still been complimented / hit on a few times.)
For the tribe, I went with the Haragin because they sound relatively normal and their legend has a call to adventure in it. Temujin set out to find the mythical land covered in grey monoliths and inhabited by fire-breathing steel demons, but has a bad sense of direction and stumbled into Eorzea during her travels before being swept up in the events of the story.
My original character and name, Cilia Grayve, was a Highlander Hyur whose name was simply pulled out of the RNG; I changed to an Au Ra at 3.0's launch. For a brief moment in Heavensward I changed to a Sea Wolf Roegadyn named Wakkahtyn Rhotarsyn ([A]woken Judge, Son of the Red Eagle; meant as a pseudonym for someone who had a past in crime but decided to dedicate his life to justice), and before that I had pretty much the same character / avatar as I do now, just with the forename "Temulun" instead. (Happily the name change alleviated the issue of being confused with the khatun of the Mol once we finally got to the Steppe.)