I found an account that the black plague gave a massive boost to Christianity in terms of the proportion of the European population which is Christian. The explanation was that Christians were generally much better at nursing, so had significantly lower mortality rate; moreover, Christians were willing to take care of non-Christians, which resulted in significant amounts of conversions.
The thing that doesn't add up for me though is that I thought Christianity was already the very dominant religion of Europe at the time of the black plague.
The argument that Christianity spread because Christians were better at caring for the sick (because they didn't run away from the dying) was made by Rodney Stark in his book, The Rise of Christianity. He was not, however, referring to the Black Death (of the 14th century). Instead, he was referring to plagues in the late 2nd century Roman empire, such as the Antonine Plague (which may have been measles or small pox -- historians aren't quite sure of the specific disease).
Stark's argument is reasonable, and many people cite him favorably, but it's a bit of a what-if speculation as we don't have reliable textual sources from this period that confirm or deny it. It's one of many explanations he suggests to explain why the rapid spread of Christianity in the early Roman empire is a perfectly normal sort of sociological phenomenon -- that larger argument is widely accepted by historians who respond to his work. I would certainly recommend the book if you want to learn more about how a small Jesus movement became, in only a few centuries, one of the most successful religions in the world.
The 14th century Black Death did, itself, create a vibrant religious response. Many Christians thought (quite reasonably!) that the world was coming to an end, and many movements embraced harsh self-denial and penitence in order to prepare for Christ's second coming. The image in Monty Python's 'Holy Grail' of monks wandering the countryside hitting themselves in the head mocks these movements, which included groups who would beat themselves to mortify the sin in their bodies and prepare their souls for heaven. You also see a great interest in death in art after the plague, as people contemplated the meaning of their mortality, and hoped for a resurrection. Europe was, however, almost entirely Christian at this point. Even its north-eastern margins, which remained pagan into the high middle ages, had been converted by the time the plague hit. The plague's effect was, therefore, to intensify the faith of many, rather than to drive pagans into the fold.