What's in a Name?

If the early church fathers were suddenly picked up out of their own times (when they were alive) and dropped into our time, what would be the first question they would ask in order to find out where the true church is in our time?

St. Cyril of Jerusalem and St. Augustine give us one possible answer to that question (embedded links and added biblical references removed):

But since the word Ecclesia is applied to different things (as also it is written of the multitude in the theatre of the Ephesians, And when he had thus spoken, he dismissed the Assembly), and since one might properly and truly say that there is a Church of evil doers, I mean the meetings of the heretics, the Marcionists and Manichees, and the rest, for this cause the Faith has securely delivered to you now the Article, And in one Holy Catholic Church; that you may avoid their wretched meetings, and ever abide with the Holy Church Catholic in which you were regenerated. And if ever you are sojourning in cities, inquire not simply where the Lord's House is (for the other sects of the profane also attempt to call their own dens houses of the Lord), nor merely where the Church is, but where is the Catholic Church. For this is the peculiar name of this Holy Church, the mother of us all, which is the spouse of our Lord Jesus Christ, the Only-begotten Son of God (for it is written, As Christ also loved the Church and gave Himself for it, and all the rest,) and is a figure and copy of Jerusalem which is above, which is free, and the mother of us all; which before was barren, but now has many children.  (St. Cyril of Jerusalem, Catechetical Lecture 18) 

For in the Catholic Church, not to speak of the purest wisdom, to the knowledge of which a few spiritual men attain in this life, so as to know it, in the scantiest measure, indeed, because they are but men, still without any uncertainty (since the rest of the multitude derive their entire security not from acuteness of intellect, but from simplicity of faith,)— not to speak of this wisdom, which you do not believe to be in the Catholic Church, there are many other things which most justly keep me in her bosom. The consent of peoples and nations keeps me in the Church; so does her authority, inaugurated by miracles, nourished by hope, enlarged by love, established by age. The succession of priests keeps me, beginning from the very seat of the Apostle Peter, to whom the Lord, after His resurrection, gave it in charge to feed His sheep, down to the present episcopate. And so, lastly, does the name itself of Catholic, which, not without reason, amid so many heresies, the Church has thus retained; so that, though all heretics wish to be called Catholics, yet when a stranger asks where the Catholic Church meets, no heretic will venture to point to his own chapel or house. Such then in number and importance are the precious ties belonging to the Christian name which keep a believer in the Catholic Church, as it is right they should, though from the slowness of our understanding, or the small attainment of our life, the truth may not yet fully disclose itself. But with you, where there is none of these things to attract or keep me, the promise of truth is the only thing that comes into play. Now if the truth is so clearly proved as to leave no possibility of doubt, it must be set before all the things that keep me in the Catholic Church; but if there is only a promise without any fulfillment, no one shall move me from the faith which binds my mind with ties so many and so strong to the Christian religion.  (St. Augustine, Against the Fundamental Epistle of Manichaeus, Chapter 4)

While the name "Christian" distinguishes the true religion from false religion, so the name "Catholic" came to distinguish orthodox Christianity from heterodox Christianity.  Interesting!