I'm 100% sure that God wasn't influenced by Hellenistic thought, Hebrew thought, or any other kind of thought. I'm saying He uses the language of the people to speak to the people. I'm not suggesting that the author believed that the Logos concept from Stoicism was factually accurate, I'm saying it was a culturally ubiquitous concept, like, say, the concepts of the subconscious, or democracy, or feminism are for us: you don't necessarily believe in them, but they are part of the culture you live in, so you understand them, and you can speak with other people about them.
I'm just suggesting that the author is using the concept of Logos to teach a universal Truth about Jesus for his culture the same way Paul in Acts 17 used the alter to the unknown god to teach the Greeks in Athens about God. Paul says, "So you are ignorant of the very thing you worship—and this is what I am going to proclaim to you." I think that's exactly what John 1:1 for Greeks; it says, you already understand that there's an active intelligence in the universe that holds all things together, now let me teach you that this intelligence was there from the beginning, was with God and was God, was made known to us and walked among us, and has a name: Jesus.
I do see what you're getting at Roads, I just don't see it applicable in John as that was directed to Jews not Gentiles. Of course what Paul said and did in Acts 17 was directed at the religious concepts of those Greeks in Athens, the center of their culture and religion. Although educated in Greek, Paul was born a Roman in Tarsus but grew up in Jerusalem and taught by Gamaliel as shown in Acts 22:3. I'm sorry but I just don't see what you see. Paul did use the situation in Acts 17 to change the Greek perception of the unknown God being their's, to the unknown God being known by the followers of Christ. I'm NOT really sure just how ubiquitous Logos really was, but obviously as far as those who were educated in Greek, there was some familiarity, and that is how Paul used it. The book of John is another story seeing as it wasn't really directed at Greek educated peoples.