I have Googled this and it seems I’ve found the name “Nellie” to be short for, basically, any other name that contains the letter “N.”
With that in mind, is there any specific name it is short for? Or is it just a sort of generic nickname for girls, like “Skip” or “Chip” is for boys?
Eleanor, sometimes.
I was going to say usually Eleanor.
Neli is a common nickname in some Slavic languages - it’s short for Anelia.
I think it used to be a nickname for either Helen or Eleanor… but over time, it became a name in its own right. Some of the more famous Nells, like Nell Cater, have “Nell” as their given names.
The most famous Nellies I can think of, however, were men. MVP second baseman Nelson “Nellie” Fox, and rapper Cornell “Nelly” Haynes.
I know one who uses it as a nickname for Ellen.
It’s a corruption of “mine Ellen”, in the same way as “Nan” derives from “mine Ann”.
I know a Janelle who uses Nellie as a nickname.
I know an Antonella who goes by Nellie, as well as a Helen who goes by Nellie.
I heard of an elephant who packed her trunk, she was called Nellie 
Around here, Nel and Nellie are usually short for Petronella, which is I believe the feminized version of Peter, or for Cornelia. However, this is almost certainly not true where you are. In English I thought it was the short form of Helen, but apparently not.
I was thinking Ellen as well, from “Nelly” Dean in Wuthering Heights who is actually Ellen.
In Polish, it’s Aniela. That was my grandmother’s name (and Nellie was her nickname). The Slavic version of “Angela”, I think.
ooooooooOOOOOOOOOOOOOOOOOOOH
You beat me to it and now I’ll have that song stuck in my head for the rest of the day.
I knew a Nell whose full name was Genelle.
For guys, it’s a variation on Nelson as for Nellie Fox, Baseball Hall of Famer. My father also had some friends named Nelson back in the '20s and '30s he called Nellie.
As do I! Hmmm…
My great grandmother was Ellen Jane Gray, who went by ‘Nellie’. I had a hell of a time finding her in my search.
My gran was known by all as Nellie, and her name was Helen. I think this is quite common in Scotland amongst that generation.