I would recommend researching some of the different types of NFP. There are groups like NaPRO, Couple to Couple League, Billings Ovulation Method, etc. If she sees that there are several versions to pick from and they are very scientifically correct, she may feel better about it. My fiance is a developmental biologist and my background is in neuroscience. Trust me, if we didn't see that there was some legit science behind it we would be VERY skeptical about the effectiveness. I actually got a medical textbook on Natural Family Planning through interlibrary loan one time. Research is key!
You wife may not realize that her doctor and the parenting magazines tout birth control's "perfect rate" in their ads. This is an effectiveness rate based on perfect use and mathematical models. It is NOT the rate she should expect for herself- but it sure sounds good in their slick and very high-budget advertising. The effectiveness rate for NFP that is usually referenced by magazines and doctors is the "actual use rate"- and a flawed actual use rate at that. Their stats generally lump all NFP together (the legitimate NFP versions and the antiquated "rhythm method" that is NOT advised by NFP teachers) and not just the more advanced ones so this automatically makes NFP look really bad statistically.
If birth control was treated the same way mathematically you would see only the actual use rates of pills, condoms and "pulling out" all lumped into one statistic labeled "birth control". We can all see that statistic wouldn't tell us anything terribly useful and it wouldn't make anyone really want to use birth control because the numbers would look bad.
Why are statistics and percentages manipulated this way? Well, NFP isn't very good for the pharmaceutical industry. Birth control is big business! They aren't going to make the competitor look good because that would mean less money in their pockets.
My fiance has told me that he is very thankful that I am also on board with wanting to use NFP. He and I agree that not only for religious reasons, but for SCIENTIFIC reasons, we cannot agree with pumping me full of hormones on a daily basis all in the name of rendering myself infertile. (The funny thing is, if any doctor asks me why I am not going to use birth control I will cite "scientific issues" just to throw them for a loop and get a fun biology throw-down going. hee hee... People spend big money to buy organic milk, meat, and other foods to avoid chemicals and hormones, yet they have no problem ingesting dangerous levels of class 1 carcinogens on a regular basis. (Estrogen above what a woman naturally produces is considered a class 1 carcinogen by the World Health Organization).
All this being said, I would suggest laying out all the concrete FACTS about NFP vs. the pill. Science is much harder to argue with than religious beliefs. Your religious beliefs are INCREDIBLY important- don't ever put them in the backseat when it comes to this- but science and facts are some of the best ways to show people WHY your religious beliefs are very well reasoned.
Best of luck! My fiance and I have had some issues with family members that think NFP is crazy, so we have had to defend it a bit lately because our families are sure that we are going to end up with too many children and broke as soon as we get married....