In Neon Genesis Evangelion, each Eva unit contains a human soul -- and this tends to be the soul of its pilot's mother. Shinji Ikari's mother Yui and Asuka Langley Sohryu's mother Kyoko are both demonstrated within the franchise to be present in their children's Evas. The case of Rei Ayanami's Eva, Unit-00, is much more ambiguous.
Rei is not a naturally-born human, but rather a clone made from the DNA of Yui Ikari and the Angel Lilith. Therefore, she does not have a mother, and her "source" Yui's soul inhabits Shinji's Unit-01. So who is the soul in Unit-00? Interestingly, evidence points toward it being Rei herself.
The Many Reis
There is not only one Rei Ayanami. Several spare Rei clones are kept in a tank, used to program the Dummy Plugs and to replace Rei if she should die. There is only one soul among the many bodies, which is stored in whichever Rei is "active". The Rei who is present throughout most of the series is the second active clone, Rei II; Rei I was murdered shortly after her creation by former NERV chief scientist Naoko Akagi, who viewed her as a rival for Gendo Ikari's affections. In Episode 23, Rei II is killed and replaced by Rei III.
There is a strong case to be made that the soul inside Unit-00 is Rei I. This is possible even though the Reis share a soul. This is demonstrated in the backstory of Asuka's mother Kyoko, who had only the "maternal" part of her soul removed and placed into Unit-02, with the rest being left in her body. Therefore, after the death of Rei I, it is possible that only part of her soul was placed into Rei II, with the other part going into Unit-00.
In Episode 14, a cross-compatibility test is held where Shinji and Rei attempt to synch with one another's Evas. Normally, a pilot cannot synch with any Eva but their own; however, if Rei I's soul is really in Unit-00, perhaps NERV thought that Shinji would be compatible with it since Rei was derived from his mother. If this were the case, it would also seem plausible that Rei might be compatible with Unit-01, which contains the soul of the woman she was cloned from.
In fact, Rei is able to sync with Unit-01. However, the cross-compatibility does not go both ways -- Shinji is unable to control Unit-00, which goes berserk. Perhaps this is because Unit-00's soul was not entirely Yui's, but also partially Lilith's.
The berserk Unit-00 begins punching a window through which Rei and the NERV scientists are observing the test. One of these is the current chief scientist, Naoko Akagi's daughter Ritsuko. While Misato Katsuragi thinks that the Eva was trying to attack Rei, Ritsuko believes herself to be the Eva's true target.
This was not the first time Unit-00 went berserk. Episode 5 previously showed that it had happened before, shortly before Shinji's arrival at NERV. During an activation test, Rei loses control of her Eva, and just as in Episode 14, it punches a window in the test room through which it is being watched by Gendo and Ritsuko. The cause of the incident is stated to be mental instability in Rei; however, just as in Episode 14, Ritsuko seems to have a different idea of what happened, which she does not share with either Misato or the audience. As chief scientist, Ritsuko would most likely know which soul was in the Eva, and as Naoko's daughter, she may also be aware of what happened to Rei I.
If Unit-00's soul is indeed that of Rei I, it has a clear motive for attempting to attack Ritsuko, who was present as an observer during both tests. Unlike her mother, who had dark hair, Ritsuko is blonde; however, in Episode 14, shots from Unit-00's point of view depict Ritsuko as dark-haired. It seems likely that Rei I's soul is mistaking Ritsuko for Naoko -- her murderer.
The Rei Inside the Eva
During the cross-compatibility test, Shinji sees a ghostly image of Rei and feels a presence he believes to be Rei trying to enter his brain. However, he seems unsure as to whether it is actually the Rei Ayanami he knows. If this presence is the soul of Rei I, it would make sense that it seems both Rei-like and unfamiliar to Shinji.
In episode 23, Rei herself refers to "the [her] inside the Eva", implying that she too is aware of a separate version of herself within Unit-oo. While Rei is aware throughout the series that her body is replaceable, it is not revealed whether she knows she herself is a replacement for Rei I. However, both she and Shinji seem aware of a presence within her Eva that is both her and not her.
When taken together, Unit-00's apparent attacks against Ritsuko, the presence both Shinji and Rei feel within it and the fact that souls can canonically be split in the universe of Evangelion all seem to point towards Rei I's soul being in Rei's Eva. Rei's confused identity as only one of many bodies is further emphasized by the possibility that she may, in fact, also be piloting a separate version of herself.