Dr. Nick: Broadly speaking each new Rei inherits the memories of the previous one (Rei III knows who Shinji is and so on), but it's unclear whether there is some information loss involved, or if this is a synching issue with the new body.
UrsusArctos: That seems debatable, since Rei 2 doesn't seem to have inherited the memories of little Rei 1 being strangled by Naoko as a child - all indications are that the memory went into Eva-00 and the Rei inside. Gendo probably backed up Rei's memories periodically, hence this incarnation doesn't consciously remember information that happened after the backup, like saving Shinji in Cut 239. However, she seems to have the soul segments from both previous incarnations, hence subconsciously remembering things that she has no conscious recollection of.
That's certainly a possibility. Perhaps memory backup is what the mystery tube is for.: