The opposite reaction is to say, "God knows His people and our human congregation is nothing, so we will not be naming ourselves anything." Invariably, outsiders will make one up for the group such as "John Smith's Church," "that former Baptist group," "the Town Hall Congregation," and so on. It is probably better to choose a reference for a group than to let others do so by default.
While the word "church" in the Bible sometimes refers to all believers everywhere, it also sometimes refers to a single, local congregation. The designations used for congregations occurs in many places in Scripture. These usually start with the word "Church," frequently add "of God" or in one case "of Christ" (Romans 16:16), and usually conclude with a preposition and place name ("at Corinth" or "of the Thessalonians"). This formula can still sometimes be used today but with two additional issues that need to be considered.
Congregational references such as "Assemblies of God", "Assemblies of Christ" and others that use common Bible words are already taken (and owned) by existing denominations. An independent group may wish to avoid those designations simply to avoid being confused with those denominations.
Believers today are much less unified than they were in the first century A.D. One independent congregation usually does not represent the entire set of converted people in a particular geographic area. For example, there may be many other believers in Seattle who do not attend "God's Congregation in Seattle."
These problems can be avoided by calling the group a "Fellowship," "Congregation" or "Meeting." Greatly limiting the geographic area can also help, such as naming a church "The Market Street Seattle Fellowship." It does not claim to define who is in the fellowship, but rather declares that certain people get together to fellowship at that place. This kind of name may sound too local or too humble, but that may be exactly what you want: a local, humble group through which God can do great things.
Something else to be avoided is using the same reference as another group, both to avoid confusion and prevent legal problems. If you include a place designation in your title, it is much easier to be know you are not duplicating someone else's name.
A little research in a local phone book, newspaper, or simple Internet search will help insure uniqueness. By not including a unique location, such as "Congregation of the Almighty God," you will have to search a much large area (e.g. a state) in order to avoid conflicting designations.
It is highly recommended that naming a church should not be based on one particular leader, either dead or alive (1Corinthians 1:11 - 17; 3:1 - 10). Additionally, do not call a congregation after a particular doctrine or practice (e.g. baptism, form of government, speaking in tongues, etc.). While many groups do this, it tends to serve as a point of division.
By placing a doctrine in the group designation, it may attract people who already believe the doctrine, but it will discourage those who do not. For example, a person who does not believe in baptism might avoid the Hill Street Baptist congregation, but may be willing to attend the Hill Street Congregation and learn about baptism when someone there teaches it to him or her.