To be very honest, there is absolutely no credible historical information relating to the cross that Jesus was crucified upon. The reason for this is that early Christians weren't really interested in any of the artifacts that would have concerned Jesus since most believed that he would be returning in the form of the second coming of Christ within their generation.

The other big challenge is that contrary to popular Western beliefs, Christianity was not a rapidly growing religion until the later half of the second century. In fact, there isn't a signal mention of Christians in pagan writing until 112 CE where (in a hand written letter) the governor of the Roman province of Bothnia-Pontos (where Turkey is today) mentions a group of people who would meet in the mornings before light, who worshiped "Christ as a god" and used to share meals together. ( Ehrman, Bart D. . Jesus: Apocalyptic Prophet of the New Millennium Oxford University (1999-07-26) Press. p. 57). In this letter they don't even mention Jesus' name since they are focusing on the actions of his follows. It's not until years later where the Roman historian Suetonius made (Suetonius, Life of Claudius. University of Chicago. http://penelope.uchicago.edu/Thayer/E/Roman/Texts/Suetonius/12Caesars/Claudius*.html) what many believe was the first indirect mention of Jesus' name in writing (this is of course apart from the Christian writers of the New Testament who had started writing about him for more than half a century prior). Because of the fact that non-jewish/non-christians of the first and second century CE weren't really interested in the progress of Christians, and due to the fact that Christians primarily wrote about Jesus' deeds, works, and messages (hence the New Testament), no one would have written about the cross until much later.

It is also worth mentioning that the Cross that the historical Jesus was crucified on was probably mass produced and mass utilized pieces of wood that would have been used to execute criminals both before and after his death, thus making finding the cross impossible to track since his disciples wouldn't have wanted to save it since they were all terrified that their leader had just been executed and went into hiding.

My graduate work is focused on early Christianity prior to the year 400, so I wouldn't be the best resource for the "lore" behind the cross, but I hope I shed some light on the historical nature of it.

Apart from the book I cited above, The Historical Jesus: The Life of a Mediterranean Jewish Peasant by John Dominic Crossan also goes into detail about the early life of Christians and you could gage from his book why something physical, like the cross would not have been any type of priority for early Christians.

I could also provide you with further scholarly reading on the topic of the Historical Jesus if you are interested.

Edits: for spelling, grammar, and clarity.