The limited liability company, or LLC, is a creature of state law. As such, it is subject to the laws of the states in which it is organized and registered. All states have some requirements for the name of an LLC and generally have searchable online databases that allow you to determine whether your name has already been registered.
Virtually all states that recognize LLCs require that the name of the business include the term "limited liability company" or some abbreviation thereof to designate its entity type. While LLC is probably the most commonly used abbreviation, "Limited Liability Co." or "Limited" may also be used in some states.
Most states have a department of corporations or its equivalent that allows you to search though its website for registered business names on file with the state. One important use of this feature is to search whether someone has already registered the name you have in mind, thus making it unavailable for use. If your name is not already registered, you should screen the database of fictitious names and DBAs -- Doing Business As registrations. You should also conduct a general Internet search to see if your business name is already in use in the state.
State law generally also requires a registered LLC name to be unique, which implicates certain features of trademark law. This generally means that you should avoid choosing a name that is likely to cause confusion in customers between your products and services and those of a competitor. Even if the exact name you want isn't the name that's registered, you could be violating someone's trademark rights if your name is too similar.
Articles of Organization
The formal document required to register your LLC with the state is usually called the Articles of Organization. One common reason these documents are rejected and the company not registered is that the name for the business is not sufficiently unique or unregistered. Thus, to save the time and money of having to refile, you should be reasonably certain by the time you submit your Articles of Organization that the name you've selected meets state requirements and is available for use.