A church group's name is important. It is something about which you should pray and seek wisdom. It is the means by which other entities will know and classify the group. The "other entities" include members of any former groups, potential new members, taxing authorities, other government entities, as well as banks and other businesses.
What a name is not
You cannot sway God's opinion of a group by calling it something like the "Church of the Most Righteous Holy Ones." It is good for each believer to seek to bear good fruit (1Corinthians 3:10 - 15), but that cannot be done on a group basis. Using a pompous title for a congregation can lead people to rely on their name, not on the fruit of the Spirit in their lives.
For we must all appear before the judgment seat of Christ, so that each one may receive what is due for what he has done in the body, whether good or evil (2Corinthians 5:10, ESV).
The opposite reaction is to be so meek that one does not even want to name their group. This humility is good, but when one is "letting their light shine," others will come to know a group by its love and ministry to others and they will refer to it as something. This writer has experienced unnamed groups such as "Jim Smith's Church," "that former Mormon group," "the Town Hall Congregation," etc. It is better to choose a designation than to let others do so by default.
Local congregations in the New Testament were not labeled by their doctrines. They are referred to as the "Church of God" with the city's name sometimes after it (1Corinthians 1:2) or the "Churches of Christ" as in Romans 16:16. Other designations include "church of the Thessalonians in God the Father and the Lord Jesus Christ" (1Thessalonians 1:1, 2Thessalonians 1:1, NKJV). Home fellowships are identified by the person who owned the house (Romans 16:5, 1Corinthians 16:19, etc.).
Since the Scripture contains no command as to "how to name a congregation" nor a consistent example, a new group should pray and use godly wisdom to determine what it should be called.
One of the most common reasons for new congregations is the break-up of an old congregation. Sometimes, the people desire a name as close as possible to what they were called in the past, even to the point of litigation.
It would be best if a name could be chosen to benefit the future, rather than the past. In situations, however, where the brethren need to heal, a designation encouraging members of former congregations can be very helpful. For example, "Nashville Congregation of Peace" might be attractive to former members of a group that broke up due to much infighting.
When a former group was too dictatorial and controlling, new congregations frequently opt for words that describe this, such as "collective," "non-denominational," "independent," "free," "fellowship," and so on.
Potential new members
How will your group best serve new people? A new congregation starting in an area where there was none before may identify itself simply by its location: "Seventh Street Community Church," "Farmville Church of God" or "Far West Fargo Fellowship."
Many believers are seeking an independent, local meeting place. If your congregation has a focus such as on prayer, praise, Bible study, local ministry or distant missions, placing one of those words in its name would be helpful to brethren seeking to benefit or serve in those areas.
Health departments, zoning boards, and county clerks (which issue "doing business as" documents) may all use a congregation's name to make certain determinations, especially if the group is not incorporated (or, in the US, does not have a 501(c)(3) designation). In some cases, using words that clearly convey a religious entity may provide certain legal protections.
In some rare cases, religious sounding designations can be a disadvantage. For example, some counties who have asserted great zoning authority have ruled that people meeting in a certain location cannot have home churches or Bible studies. The brethren, in one such county, had to call their meeting a "party" to continue meeting.
If your local government is not known for this kind of trouble, then this does not need to be a factor in your name choice. If there are neighbors that might be impacted by your regular meetings, be a good neighbor and try to make sure that your attendees do not occupy their street parking, walk across their property, and so on. It is better to avoid complaints than fight them.
Do not choose a name that will be confused with another group. You are better off, in the long run, by choosing a unique designation for your new church or fellowship.
You can apply, in America, for a trademark or service mark, which gives you national control of your name, but that is not an absolute guarantee of preventing litigation. Someone can always legally challenge your trademark. If your mission is serving God (not marketing or money), it is almost usually cheaper and easier to alter your group's name than it is to litigate about it. Pray, consider these points, and get busy!