Why did Christianity in Ancient Rome attract so many diverse people?

Answer:

The official or state religion of Rome was built around a modification of the Greek Olympic pantheon, with the add-ons of household and hearth gods plus hidden spirits (lares). Cicero described the official religion as a compact with the gods - give them due honour and they would look after the state. With no provision for an afterlife in the official and private gods, there was a gap to be filled as people simply didn't like the idea of simply being snuffed out, so mystery cults arose to accommodate this.

These innumerable mystery cults included one called Christianity. The popularity of the mystery cults was that initiates, through personal engagement with the god through the rituals of the cult, were given the keys to an afterlife, as opposed to the nothingness of Hades (or Sheol to the Jews) which was just a repository for the shades of the dead - a blotting paper).

The early mystery cults had single cult centres, distributed mainly in the eastern Mediterranean (Dordona, Eleusis, Samothrace, Alexandria etc) and it was expensive in time and money to travel to them and pay the fees to become an initiate, so they remained the province of the few. Then some cults travelled to the customer, which meant that a much broader range of people could join. These popular mobile cults included Mithras, Christianity, Isis, Serapis etc. The Christian cults admitted all comers, even women and slaves, at no cost, available in all towns, and meeting in private houses which avoided the need for temples or the cost of sacrifices, which gave it a great advantage over the others to potential mass enrollment.

Emperor Constantine was an initiate of Mithras, however he was looking for a unifying religion to parallel and work in with his political government, and as Christianity had developed an hierarchical organisation of bishops to organise and control it, which the other cults didn't, it was politically more useful to connect with it rather than Mithraism, and organise a merger of the Imperial Cult of Sol Invictus with Christianity (from the Sun came the halos which began to appear around Christian heads). Its chief competitor as top mystery cult, Mithraism, was then put down over the following century by the ascendant Christians, its priests killed and underground cult centres sealed off, leaving the field free for the Christian authorities to make and coerce conversions and as sects within it emerged, to fight it out amongst themselves.