They would have been Nestorian Christians. I'm having trouble locating the source, but have read that Christianity thrived in Bahrain during the Sassanid era (when Bahrain was within the Sassanid Persian empire) in part because of its fringe location. Christians were treated as potential fifth columns to Rome (after the empire's fourth century conversion to Christianity), and it could act as place of refuge. (EDIT: The above portion misrepresents religion in east Arabia in that period, see CptBuck and shlin28's comments beneath!) We know that a Christian monastery existed on the Bahrain islands, as it gives its name ("Al Deir", literally "The monastery") to a village on the north coast of Muharraq island.

Though Bahrain's pagans converted en masse to Islam during the life of the Prophet, many Christians and Jews did not. Note that Bahrain in this period refers to the greater region, stretching roughly from Basra to the borders of Oman along the east Arabian coastline. Probably, as with almost all the territories conquered/consolidated in the early Islamic era (up to the end of the Umayyad Caliphate in 750 AD), conversion to Islam from Christianity and Judaism was gradual. Even so, there's nothing to suggest that they constituted more than a minority of the population during the periods both immediately before Islam and after conversion to Islam.

Christianity certainly died out, though exactly when we cannot say. Recent archaeological digs have unearthed pig bones, stripped for meat in a butcher's fashion, and dating to the 10th-13th centuries (the articles available to me right now do not make it clear to which century, but it was to the approximate period).

Since pigs are not native to Bahrain, that suggests that someone imported at least one pig and slaughtered it. We cannot derive a conclusion from this finding, only a number of possibilities, one of which is that communities for whom pork is not forbidden (Christians most obviously) lived in Bahrain at that time.

It could alternatively suggest unorthodox Muslim society. Dog bones were also found in a nearby site, with marks of butchery for possible human consumption (also forbidden in Islam). So again it could suggest unorthodox Muslims. But it may also be a sign of famine, starvation and desperation. These findings were made in one of the major medieval urban centres on the Bahrain/Awal island, so perhaps it reflects a cosmopolitan demographic at that urban centre.

We simply don't have enough information to say anything more than that Christians may have lived in Bahrain at that time. They would have been a minority.