Your name must be unique, and not deceptively similar, to any other trademarked name or business. It is also required that your name not be used to intentionally misrepresent the products or services you offer. For LLCs, nearly all states will also require you to add a signifier of your limited liability status, such as "LLC" or "L.L.C." to the end of your company's name. You may be able to operate under a name other than your formal LLC name by applying for and using a dba.
Outside of the normal statement filing season, you may fill out a Statement of Change form with the date you went out of business, the status and disposition of any equipment owned or used by you at the time the business closed. If any of the property was sold to another person or business, please indicate the buyer's name and address. Make sure to identify any property that reverted to your own personal use as household personal property.
Where the equipment you use in your business was acquired as a gift, you may report your estimate of its current market value on the Business Property Statement (that is, what you think it would sell for in the open market place). Enter that estimated value in the most current year's cost line and add a note indicating that the entry is an estimate.