It would be a basic biologic protection system. The "core" would need to produce a sort of totipotent stem cells and, probably, keep the blood saturated in them. The stimulus to produce such cells would then be the depletion of such cells in the blood.

At the same time the organ has to "retire" the cells grown too old and weak, and implement some kind of regenerative check on all cell lines to prevent random mutations from devolving into cancers (see "integrity check", below). And/Or maybe the Hayflick limit for "normal" or fully grown cells is much lower, so they don't have the time to degenerate but rather go into apoptosis - they live fast, die early and leave a beautiful corpse.

(So another difference of the Evas would need to be that their cells are on average much "younger" than a normal human's).

If we make this work this way, at least two awkward limitations follow.

One: reproduction becomes incredibly complicated, unless the core allows for a "grace period" before going active (maybe just after puberty?) and pregnancy shuts down the core enough to let a foetus grow and mature to term undisturbed (a newborn will have a DNA which isn't the same as the parent's, and the DNA check routine would kill it at a very early stage).

Two: the core itself cannot "self-check". Immature core cells are immature core cells, mature core cells are totipotent stem cells. In some ways, the core is a stabilized tumour, and in some other ways it behaves like a liver. It can have some limited regeneration capabilities, but the reproduction of immature core cells is a very, very touchy business - triggering it too often or too fast is likely to make the whole process go awry in some terrifying way. Most likely, the core goes fully cancerous and starts eating up the rest of the organism.

(It might be possible to have an Evangelion regenerate by pumping him full of some metabolism-slowing drug, and keeping him hypothermic and in a coma, so the core doesn't exceed its design specs. Or transplanting a large enough compatible core chunk).

Hypothetical core development

• at birth: the core begins to grow (e.g. somewhere under the heart).
• age three to fifteen: the core grows but is otherwise nonfunctional. Normal body growth can take place, bones are digested and remade longer, new organs can develop and mature.
• age fifteen to eighteen (more or less): the core activates and takes some roles of a human's Major Histocompatibility Complex, plus some really wicked, DNA-based integrity check system. Some biological effects and possibly a shock occurs. Also, any damaged or mutated tissues are swiftly killed and replaced, possibly triggering some form of septic shock or rhabdomyolisis. Bottom line: not all Evas survive their coming of age.
• age eighteen onwards: the core has the blood saturated in "repair workers". Most large scale cellular damage can be repaired. Actually, large damage to the brain will be repaired but leave the victim with severe mental effects, amnesia being the least.
• during pregnancy, pregnancy hormones shut down the core and floating cells so that they don't "think" it's a good idea to do something for that poor swollen uterus full of fluids and who-knows-what. Downside: a pregnant Eva is much more vulnerable than a human, since it has next to no regenerative powers (they'd be superfluous with a working core). As a stopgap measure, a wounded pregnant Eva might immediately abort the pregnancy and return to "full defense mode".
• normally, the core awakens periodically and tops up the reservoirs at leisure, and it always filters out weakened stem cells from the bloodstream, destroying them and using the raw materials to build new cells.
• a large systemic shock (e.g. loss of a limb) sends the Eva in a regenerative coma, with metabolism reduced to the bare minimum and just enough circulation to keep repairs going. The core goes into low-level, long-term activation and wears out a bit, eating into its own regenerative reservoirs. It's important to keep the Eva well fed, otherwise his chances on recovery depend on body fat and muscle reserves. In ancient times, fat was synonym of health; for Evas it is still so.
• in case of multiple, repeated activations, while the rest of the body gets "iguana-like" regeneration, the core itself has "lizard-like" regeneration, and the mechanism goes awry. As a result, the Evangelion dies, not too horribly since the heart is the first to go.
• lower genetic diversity between Evas allow easier core transplants between them.