Theory and Analysis:Eva-03's Soul

Toji's Mother

Although it is never directly stated whose soul resides in Eva-03, it is likely Toji's mother's. The soul of Shinji Ikari's mother, Yui Ikari, resides in Evangelion Unit-01, and the soul of Asuka Langley Soryu's mother, Kyoko Zeppelin Soryu, resides in Eva-02. It is stated that Shinji's school class is actually a secret pool of potential Eva pilots, all grouped into one place. Also, it is gradually revealed or implied over the course of the series the mothers of all of Shinji's classmates are dead; (Kensuke Aida directly states this, and Hikari Horaki and Toji also heavily imply this). Thus it seems to be implied that there is some connection between all of the potential pilots having "dead" mothers.

Then, consider this, from the Classified Information:

A unit called a core is used to eliminate the discrepancies in Eva, a human being with the body of a god and a soul. The soul of the pilot's mother, which is placed within the core, acts as a medium and makes piloting possible.

Due to difficulties encountered in the development of Eva-00, the core's system is poorly constructed, and she falls behind in performance capacity as a result. Another reason for her inferiority is that Eva-00's pilot, Rei, does not have a mother."

This makes it plain that with the exception of Rei, the pilot's mother is always chosen as the Evangelion's soul.

Following this chain of evidence, the almost certain conclusion is that Evangelion Unit-03 contains the soul of Toji's mother.

Alternate Theories

Toji's Sister

There are some who hold that soul was that of Toji's unnamed sister, finding sinister undertones in Ritsuko's comment: "Getting his sister transferred to our hospital here at Headquarters was the one condition he gave”, the sister's residence inside a Nerv hospital, and absence of any indication that the sister was conscious and sane (as opposed to having lost her soul to an Eva contact experiment, leaving a living body behind, like Kyoko and Unit-02). Proponents of this theory dispute the "souls in storage" theory, and believe that the death of Toji's mother at some point in the past rules her out as the soul. Proponents of the "sister" theory also question the canonicity of the Classified Information, as it came from a game, dismissing its statement that Rei is the only exception to the "mother's soul" rule.

Proponents of the "mother" theory counter that Ritsuko seems to be saying that moving his sister to a Nerv facility was a condition that Toji imposed upon them, not something they suggested. They see no sinister undertones here at all, and just see it as Toji still prioritizing his sister's care, as he did when he missed many days of school earlier in the series. They also point out that there is no evidence that his sister isn't conscious and sane. For instance, in Episode 19, the injured Toji asks Hikari to visit her for him. This is not followed up in any way, for instance by a scene of hospital officials giving Hikari the run around.

Another Rei/Lilith Fragment

When Kazuya Tsurumaki was asked at Otakon 2001 whose soul was in Unit-03, he and his translator evinced extreme confusion and he ultimately offered the suggestion that it might have had something to do with Rei.

Due to the seeming confusion, many people take this statement with a grain of salt.

In Rebuild

It's not been made clear if the soul mechanics in Rebuild match those of the original series, but in 2.0, Toji's sister is pointedly shown alive and well while being released from the hospital before Asuka volunteered for the test of Eva-03, and despite having similar colors to Toji's jacket, Eva-03 is not connected to Toji at all. Thus the "sister's soul" theory is invalid in Rebuild, and the "mother's soul" theory is severely in doubt as well. The situation in 2.0 is ambiguous (the casual choice between Rei and Asuka offered to Misato suggesting that the soul did not matter at all).