Denominations and man-made names aside. Is Church of Christ the only biblical name that can be used on a church sign?
First, "church of Christ" is not the "name" of the church, but rather is the description of the church. It is the church that belongs to Christ.
Second, as to your question of whether "church of Christ" must be the "name" on the church sign, the answer is no. There are several descriptive terms used of the church in the New Testament. One is just as scriptural as the other. They are:
church of God, Acts 20:28; 1 Cor. 1:2; 1 Cor. 10:32; 1 Cor. 11:22; 1 Tim. 3:5;
house of God, church of the living God, 1 Tim. 3:15;
church of the firstborn, Heb. 12:23;
churches of Christ, Rom. 16:16, speaking of a plural number of congregations;
churches of God, 1 Thess. 2:14; 2 Thess. 1:4. speaking of a plural number of congregations.
Sometimes the church is simply called “the church,” or “the body, which is the church.” This was sufficient because there were no denominations using man-made names or descriptions. The term “church of Christ” came to be the term applied to New Testament congregations that sought to restore New Testament faith and practice. A denomination adopted the scriptural term “church of God”; however, it takes more than a descriptive term that is scriptural to be a New Testament church, and that includes the term “church of Christ.” There are congregations that have “church of Christ” on their sign that have departed from New Testament faith and practice. The name of the sign will not save them, or anyone else for that matter. The saved are those who do the will of the Father, including that which is on the sign. Matthew 7:21-27.
You Must Hear the Gospel
You Must Believe
You Must Repent
You Must Confess
You Must Be Baptized
You Must Be Faithful Unto Death