Partially quoting myself from an answer to a similar question

Firstly, martyrdom tends to have a pretty powerful effect, regardless of other factors. The death of a beloved leader might just be the kind of spark capable of initiating the kind of spread that we see in Christianity.

Beyond that though, there was a lot going for Christianity that allowed it to spread beyond what that initial spark started. Christianity in general was incredibly accessible to people of low social rank, which is incredibly important for the rapid spread of religion. Unlike many mystery cults that had fees to join, Christianity was free to join and actually encouraged its members to support each other and to support those in need. This allowed it to spread rapidly amongst the lower classes; even if someone originally came to their meetings for the free food, eventually they might come to believe the message. Beyond that, it had the very enticing message that there was a better life after the current one. For slaves and servants this would be incredibly attractive. It also had all the allure of other 'foreign' religions (like the cult of Isis or Mithras or even Judaism itself) but none of the drawbacks (fees, difficult to follow rules, edicts of cleanliness).

Further, I think a case could be made that it was Paul, not Jesus, that made Christianity so successful on the large scale.

Paul's particular brand of Christianity had a lot going for it that allowed it to be even more successful. First, Paul spread the religion to the gentiles (non-Jews), which opened it up to way more people than if it remained exclusively/primarily Jewish. Paul also preached that gentiles did not need to abide by the Jewish Laws. This meant no circumcision (quite the hurdle for new initiates) and no need to change one's diet.

It's also important to note that Paul was, in a sense, not really preaching about Jesus - the carpenter from Nazareth - but about the Christ. This distinction is important, especially considering that the Christ-centric (as opposed to Jesus-centric) view of Christianity was really what became the Christianity that still exists today. It could almost be imagined that, even if Paul had chosen a different apocalyptic preacher to be his Christ, his religion would still have become the dominant force that it did.

So, martyrdom, plus enticing message, plus accessibility might be part of the recipe for a prosperous religion.

I won't speak as one having authority, but I'd like to guide this discussion in a direction that interests me.

The short answer (being coopted into the Roman State religion) doesn't answer how this unpopular Jewish sect lasted long enough for that to happen.

What the New Testament says about this is interesting. According to the account of the New Testament, the disciples were dispirited and disunited after Jesus' murder. Jesus' resurrection and subsequent 40 day ministry with his disciples had an effect on them, and turned the listless bunch into dynamic proselytizers who now had license to preach to non-Jews, and establish communities outside of Israel.