With revelation in chapter 507, we can finally eliminate the mother/daughter theory about Eileen and Erza. With the new information, here's my theory:
#1 Eileen was accidentally created by the R system back when Erza was trapped in the lacrima. The R system was meant to be for resurrecting the dead. It required a living sacrifice. Perhaps Jellal was mistaken as to how it actually functions, it did not work because Zeref was never dead, or it malfunctioned when Natsu destroyed it.
Either way, the R system ended up creating a duplicate of Erza instead (Eileen). The 8 years of time skip would explain Eileen's older appearance and her greater magical power. Also recall that the explosion of the R system was much smaller than anticipated. At the time, Erza thought that Jellal was responsible for containing the explosion. But now, another possibility is that all that magic power was absorbed into Eileen, which explains why she has such tremendous magic power.
Finally, this theory would also explain why Eileen wanted to kill Erza, because as long as Erza is alive, Eileen would remain a copy of the original. No matter how powerful Eileen is, she will always remain a copy, something inferior, so that is why she wanted Erza dead so that there is only one of them.
#2 Eileen created Erza from one of her organs (hence the scar on her abdomen) using her enchantment magic. It would be similar to the biblical story of Eve created from Adam's rib. Since it came from her, it is superior to her other enchantments made from inanimate objects. Somehow they both got separated in the past. We will find out soon why Eileen hates Erza so much that she wants her dead.
#3 They are both clones created by another person. Zeref's experiment? God Anhkseram?
#4 They are both twin sisters.
All four theories above would explain Eileen's cryptic speech that they are both the same person. From the very beginning of Fairy Tail, there has been zero mention of Erza's biological parents, if she even have any! The mystery will finally be solved soon. I really hope it isn't another BS parallel universe or time travel explanation though; those have been done to death.