This is a tough one. This is an interesting position Apostle Paul had taken. He argued that there's much at stake here. Even our rights can and should be sacrificed for the greater good of the Name of Christ. Remember, this is said in the context of much disunity in the Church and other compromises mentioned in Paul's letter. The Corinthians have become very carnal and were going back to their old ways. They forgot their position in Christ, as Christians. There were lawsuits taken against each other. Paul sarcastically said that even the least in the church ought to be able to judge the case, than bringing it to pagans.
Paul took such a strong position that he went on to say "Why not rather be wronged? Why not rather be cheated?"
This is radical and if we understand this in the context of the previous post I had on being a willing slave, how then do we die to our rights? If we have died to ourself and live for Christ, dead men have no rights! Isn't that's what missionaries do, dying to their rights for a comfortable life and moving to foreign culture for the sake for the Gospel? Isn't that's what many in our churches who give up what is their right to a comfortable private life, give it up instead to serve others regularly, within or without the church?
I think Paul's hitting the nail on the head, by basically saying that the heart of the problem is the problem of the human heart. We are inherently SELF-CENTRED!
1 Corinthians 6:1-8
1If any of you has a dispute with another, dare he take it before the ungodly for judgment instead of before the saints? 2Do you not know that the saints will judge the world? And if you are to judge the world, are you not competent to judge trivial cases? 3Do you not know that we will judge angels? How much more the things of this life! 4Therefore, if you have disputes about such matters, appoint as judges even men of little account in the church! 5I say this to shame you. Is it possible that there is nobody among you wise enough to judge a dispute between believers? 6But instead, one brother goes to law against another—and this in front of unbelievers!
7The very fact that you have lawsuits among you means you have been completely defeated already. Why not rather be wronged? Why not rather be cheated? 8Instead, you yourselves cheat and do wrong, and you do this to your brothers.