You beat me to it. My ancient history course and my Latin are rusty, after 60 years with little occasion for use. I think the story, etc. is this. The Emperor Constantine's mother, Sophia, became a Christian. (It is she who went to Jerusalem in search of relics, and returned with a relic of the Holy Cross according to tradition). And it is she for whom the Basillica of St. Sophia in Constantinople was named (now called Hagga Sophie, or something like that since it is now a mosque in Instanbul). She pressed Christianity on her son, Constantine. He converted after seeing a cross in the sky, and said in this sign we will conquer. Thereafter the Roman Empire shifted from various cults to Christianity.
I believe Constantine the Great's mother was Helen of the Cross, not someone named Sophia. After her divorce from Constatine's father, she devoted the rest of her life to religious pilgrimages. She was the founder of the Church of the Holy Sepulcher and the Church of the Nativity. According to some legends, she was the discoverer of the True Cross in Palestine.
Sophia refers to "Divine Wisdom" & the current Hagga Sophie's building is credited to Justinian.