Kenneth Scott LaTourette, church historian, writes about the success of Christians in the early centuries...
Inevitably the question arises: Why, from being the faith of a small, persecuted minority in competition with other religions which appeared to have better prospects of success, did Christianity eventually enroll the large majority of the population of the Roman Empire? To that outcome several factors contributed. In the disintegration of the existing order which by the end of the second century was becoming obvious many individuals were seeking spiritual and material security and believed that they could find it in the Christian faith.
By the end of the third century, while enlisting only a minority, the Church was Empire-wide, was more comprehensive than any institution except the state, and gave to its members a sense of brotherhood and solidarity. Christianity assured its adherents what many in the ancient world were craving -- high ethical standards, a spiritual dynamic in which was power to approximate to those standards, and immortality. The Church was inclusive: its brotherhood included both sexes, rich and poor, intelligentsia and men and women of no intellectual attainments. Many intellectuals, including Augustine, found in the faith not only moral power but also, in the incarnation, the Word become flesh, what was absent in the highest philosophies of the time. The constancy of the martyrs awakened the admiration of thousands. So did the fact that Christianity was uncompromising in its demands. One modern scholar, T. R. Glover of Cambridge University declared that the Christians out-thought, out-lived, and out-died the adherents of the non-Christian religions.
The primary source of the appeal of Christianity was Jesus -- His incarnation, His life, His crucifixion, and His resurrection. Here was the sense of security and of meaning in a perplexing universe and in a society whose foundations were crumbling. Here were the command for and, although imperfect, the realization of a comprehensive fellowship. Here were high and exacting ethical commands and the proved power to approximate to them. Here was victory through apparent defeat. Here was the certainty of immortality in ever-growing and never-ending fellowship with the eternal God Who so loved that He had given Himself in His Son.
From Chapter 4: "The Initial Five Centuries of Christianity" in Christianity Through the Ages by Kenneth Scott Latourette. The wall painting of the Lord Jesus, Alpha (A) and Omega (W), above is from the 4th century tomb of Commodilla, south of Rome.