I think she is not erza sister or her mother. Mishma already trolled us twice with chapters name.
now I am about to release the most bs crackpot theory ever, and yes I have heard of the kidney one too.
What if Eileen is a springgan.

No no
I mean an actual springgan.
Look around, we have dragons, demons, gods and mages. But no fairy. And this series title is fairy tail.
We already have a dragon. natsu
And we are about to get a demon. Gray.
I am expecting Lucy to master one magic. Mage.
And we are left with one main character. Who is nothing. And have no relation to any of the big three entities we have in this series (end zeref and aconologia)
So is erza going to become completely irrelevant? Mishma favorite girl. Seems hard to believe
The way I see it, whenever natsu wins a battle, a dragon is depicted. He represent a dragon
Whenever erza wins, we get to hear about fairy tail guild. Erza embodies Nakama speeches even more than natsu.
Erza is already known as Tatiana, the queen of fairies. A fairy
So what if erza and Eileen are two parts of one fairy. Destined to combine to become the legendary fairy.
Remember when aconologia was asking Eileen who exactly she was. What was he asking? He already knew she was zeref springgan. Why did he want to know her name.
Could it possibly be that he thinks what she did, human couldn't possibly do?
And what with those funny eyes? It doesn't look like enchanting or attaching something, things which we know Eileen is master of.
This could be an easy way for mishma to keep erza relevant without attaching her to END (gray), zeref (jellal lucy), aconologia (dragon slayers) and one magic (lucy)
And its kinda how erza is always shown.