I wasn’t aware of a different name for the Eastern Church.
Yea, it's one of those things that only a religious nerd might know about, (such as the Pope;)
Berenike is the Greek for "bringer of victory". Veronika (verus - Latin="truth" and Ikon - Gr. = "image") makes her name "true image".
A wide tradition (small "t") in both the East and West has her as the woman who had a 12 year blood flow healed by Jesus in the synoptic Gospels. If her given name for Greek was Bernice, then Veronika would be a title given her because of the event meditated on in the 6th station of the Cross.
There is, of course, a whole story about her life (getting married to a Christian convert, carrying the veil around, etc.) post Resurrection of Jesus.
Christ imprinted His image in her heart before all else and she freely transformed her life to correspond to His Grace. That really is all that is important for us to reflect upon. Is His image imprinted in our hearts as well?