Why did Darkness pick the name "Darkness"? We understand she's embarrassed about her real name Lalatina and noble status. But was it ever explained how/why she specifically came to choose to go by Darkness of all possible names?
There is nothing confirmed, but there are a lot of possibilities.


Darkness is a status effect in Final Fantasy, which is also known as Blind. It makes the affected miss their attacks. So her nickname could be a reference to this.
It could refer to her masochism, her inner "Darkness"
It could be a slightly modified Dustiness. 
Dustiness in Japanese is ダスティネス. Darkness is ダクネス.


Keeping the first and the last two katakana of Dustiness, changing the "su" ス into a similar looking "ku" ク and discarding the other two katakana, and you have Darkness.