There are plenty of reasons why Christianity became so popular in Rome.
First, the eastern part of the Roman Empire consisted of many civilizations, many that were older than Rome and had their own religions. After conquering these lands Romans appreciated and valued these eastern religions more than their own and started using them themselves, ditching the old Roman gods, although, there were Romans who valued the old gods highly and shunned the new gods. The old Roman gods were more worshipped in the countryside, the new gods in the cities where traders from the east came to.
In the Second Punic War, the worshipping of Cybele (originated from Phrygia in Asia-Minor) was quite popular in Rome. Near the end of the republic, the worshipping of the Egyptian god Isis became quite popular. I am not sure why Romans stopped worshipping these religions, so I will leave that. Although, I think it is simply because of Christianity and how it came to be more popular than these other religions.
Quickly after the Roman Empire came to existence (that is, emperors started ruling) the worshipping of Persian god Mithra spread through the empire. This cult of Mithra was the biggest competitor of Christianity. The main reason for the downfall of Mithra was that women and slaves were not allowed to worship him.
Christianity originally arose as a branch of Judaism. Jesus started spreading his new message and gained quite a few followers, mainly because of his moral message, the selflessness and love for one's neighbour. After his death, the apostles started preaching Jesus's teachings and convincing people (almost entirely Jews), that Jesus had been the Messiah. Paul the Apostle had the opinion that everyone should be able to believe in the message of Jesus and believe in God. He removed the requirement of circumcision and more so more people would convert to this religion. He went on missionary trips to Asia-Minor, Greece and eventually Rome. At this point, he had changed this new branch of Judaism enough that it could be called a separate religion.
Roman authorities had generally tolerated Judaism because the Jews were few, they reacted badly when religiously provoked and Judaism was not meant for spreading and converting other citizens of Rome. Christianity on the other hand was quickly spreading, much to the dislike of the authorities because everyone could convert and Christianity forbade the worshipping of the old gods, as well as the emperor (now it was common that the emperor was worshipped as a god, especially in the eastern part of the empire).
Organized persecutions against Christians first began after The Burning of Rome, 64 AD. Nero needed scapegoats and Christians were perfect for the part. At this point Romans thought Christianity was just a branch of Judaism. In the last decades of the first century AD, the New Testament was brought to written form for the first time, in Greek. (Worth mentioning that the word Christ comes from the Greek word for Messiah, Jesus is the Greek version of Joshua; many common names today in Christianity come from this translation). Around 100 AD Christians started organizing their assemblies.
In the early ages of the Empire many educated Romans had dropped religion entirely and took up philosophy, these educated people were more susceptible towards the monotheism of Christianity. The eastern religions often had deep emphasis on religious empathy that people from the Western part of Rome often sought after. While many religions tolerated each other, Christianity did not. Christians believed that their god was the only one and shunned and persecuted other religions. Christianity was also young and therefore more adaptable then the rest. Poorer people were often attracted to Christianity because Christianity was supposed to make sure that the poor could survive when hard times were at hand. Like I mentioned before, Christianity also allowed slaves and women to practice the religion, but they were often not allowed to practice the others. Therefore, in a way, Christianity became a shelter for the subdued and oppressed, and in the Roman Empire, many people were subdued and oppressed.
Eventually the Roman authorities realized that Christianity could not be beaten with persecutions and in 313 AD, pay in mind that emperor Diocletian had held the biggest persecutions against Christians in history around 300 AD, Christians were given complete religious freedom by law. This law came from Constantine and he gave Christians many benefits, such as jurisdiction over their own affairs. Around 300 AC, it is believed, that almost half of the population in the eastern part of Rome was Christian and one-eighth of the population in the western part was Christian. The church had gotten a lot stronger in turmoil of the third century AD and was at that point one of the best-organized institutions in the entire empire. Constantine has probably considered Christianity to be a possible symbol of unification in an ever-dividing empire.
After religious freedom was given Christianity strengthened a lot and almost every emperor, succeeding Constantine was Christian. In 380 AD, emperor Theodosius made Christianity the state religion, and in 391 - 392 AD, he banned the worshipping of all other gods. Christianity had won its final big victory.
Please correct me if I am wrong about any of this, and if I am, then I apologize. I also hope my grammar and overall English usage is acceptable, and most of all readable. I do also realize that I may have talked too much about the history of Christianity, but I feel it is too connected to the way Christianity became so popular.