If by the Byzantine cross you're referring to the cross used in the Eastern Orthodox Church, then it came from an interpretation of the bible and the cross on which Jesus was said to be crucified.

John 19:21 states that Pilate ordered a plaque to be nailed to the cross with the letters INRI above Jesus' head. INRI stands for Jesus of Nazareth, King of the Jews in Latin (according to the Bible the plaque was written in three languages, Hebrew, Latin, and Greek).

The slanted bar at the bottom is something that crosses would have had, to make holding a body up possible.

Sources: the Bible, http://en.wikipedia.org/wiki/Jesus,_King_of_the_Jews