I suspect we've all known people like this, and it's admittedly hard to get along with them (unless you always let them have their own way, which may cause even more problems). In reality, however, we all can be stubborn from time to time, and we need to recognize it and ask God to help us overcome it.
Why do some people always have to be right, no matter what the actual situation? One reason is pride -- an attitude that says, "I know more than you do, and therefore I am right and you are wrong." But pride is a dangerous thing, because it blinds us to our own faults and cuts us off from others, and from God. The Bible warns, "I will break down your stubborn pride and make the sky above you like iron" (Leviticus 26:19).
But I've found also that stubborn people are often very insecure; they're afraid people will look down on them if they admit they might be wrong. But this isn't true; most people actually admire someone who's humble and can admit they don't have all the answers.
Pray for your sister and her husband. Pray not only for their relationship, but that he will turn to God and humble himself before Christ -- who humbled Himself for us by dying on the cross for our sins. The Bible says, "Be completely humble and gentle; be patient, bearing with one another in love" (Ephesians 4:2).