Because this is a vast topic, I can touch only briefly upon the historical development of the idea of Satan which comes long before the early Christians. Although the Old Testament does not clearly tell us about Satan's nature nor his role, there seems to be a development of thought about him. And the name itself comes to us from the Old Testament.
Sometimes, this word meant simply to oppose or harass as in the story Numbers 22:22) of the angel who prevented Balaam's donkey from going where God had told Balaam not to go. At other times, Satan is the tempter of humans and the adversary of Israel and of God. The first reference to Satan as a demonic spirit: "Satan stood up against Israel" (Chronicles 21:1) contrasts with reference to the same incident: "The anger of the Lord was kindled against Israel" (2 Samuel 24:1).
Adversary of humans
Used in a legal sense, it meant prosecutor or accuser as in Zechariah 3:1-5 and especially in the Book of Job where Satan is the public prosecutor of heaven. He keeps watch over human beings, but he also tempts them to rebel against God. Therefore, he is an adversary of humans. He is not on an equal footing with God but a creature.
In the New Testament, Christ came "to destroy the work of Satan" (1 John 3:8) who is given various names: the evil one and Beelzebub (Mark 3:22), the enemy (Matthew 13:39), the adversary
(1 Peter 5:8), the accuser (Revelation 12:10). Satan is the chief of demons, his kingdom is single and united Mark 3:22-27). The letters of Paul bring out the close connection between Satan and sin.
The early Christian thinkers simply reflected on Satan as they did on other aspects of belief. In the beginning of the second century, there were the apostolic fathers. Clement, bishop of Rome (94-97) seemed to see the devil as a distinct personality. Ignatius calls the devil the ruler of this age whose purpose is to thwart Christ's work of salvation. Although pitting himself against each individual, Satan's power and knowledge are limited by God. The Shepherd of Hermes (written about 140) stressed the struggle between good and evil spirits within the human heart.
In the latter part of the second century were the apologetic fathers, among them Justin, Irenaeus and Tertullian. Realizing that Christianity had to compete intellectually with Greek and Jewish thought, they sought to apply analytical and logical reflection to revelation.
Justin (c.160-c.165), was influential as he was the first to discuss evil in theological terms. He believed the absolute inferiority of the devil to God and that Christ has broken the power of Satan and will fully destroy him at the Second Coming.
Irenaeus (c.140-202) was more concerned with the alienation of humanity from God than with the mythology of demons. He emphasized human responsibility for sin as the devil, in spite of his limited power, tries to deceive humans. Tertullian's (170-220) diabology was even more influential. Because of the gnostic belief of the evilness of matter, he stressed that evil consists in attachment to the things of the world and not in creation which is good. God made Satan for glory but Satan made a free choice for evil and now tries to pervert God's creation.
Fourth and fifth century thinkers, such as Jerome, Ambrose and Augustine, spent more time and effort on basic Christian doctrines. Therefore, reflection about the devil was more a refinement of previous thought.
Questioning the meaning and cause of evil in the world, they responded that evil is the choice to sin when free will is exercised. Augustine believed the devil can never repent but Augustine couldn't figure out why not.
Therefore, it is clear that Justin and the other early Christian thinkers tried to analyze and put into words what they understood about the devil.
They didn't really "invent" anything as their ideas built upon the Bible and Christian tradition.