You won't have to wait any longer folks, I am Jieng/Dinka like her. And a nerd while at it.
The correct spelling is actually Athieng Mou. Athing Mu is an American corrupted spelling of Athieng Mou. Her names start like this, Athieng Deng Tong Mawien Mou. The spelling in Dinka language (Thongjieng) is: Athiɛ̈ŋ Deŋ Tɔŋ Mawiɛ̈n Mɔ̈u.
Here are the meanings:
Athieng in Dinka refers to a cow with a grey-brown shade. This cow is named as such because those colours resemble the colours of a Tiang antelope ( Google Tiang). Tiang is so typical to South Sudan that the Dinka word, Thiang, is the scientific name for this specifies of Antelope. Google Tiang migration. Athieng simply means Tiang-coloured cow. Her dad either paid an Athieng cow as part of his dowry or if she was named after one of her grandmothers, then one of her grandfathers down the lineage had an Athieng cow as part of his dowry.
Mou literally means alcohol, a type that is specifically brewed traditionally in Dinkaland. There is a specific story to that too.
Deng means rain and it is a name that is passed on from grandfathers.
Tong means - war or spear
Mawien comes from wien which is a rope used to tie down cows in a cattle camp (see images of Dinka cattle camps on Google)
A big chunk of Dinka names are derived from cow colors or interrelation of colors on a cow. And those color names are in turn derived from animals that have similar shades.
Note that I explained just her first 5 names, the concept of fixed last names does not exist in our culture. Your names are a chain of your ancestors' names. Her full name is more than 10 names-long (my count goes to 14). But in the West, we are forced to adapt to Western last name conventions. Therefore, she is using Mou as her last name but it is actually the first name of her great great grandfather. We are patrilineal.