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.
For businesses in industries like construction or real estate, where unforeseen circumstances and hazardous conditions may hold the owner responsible, consider starting an LLC. The protection gained means you will not be held personally liable, protecting you and your family from litigation or the debts of the business. An LLC may not be the best choice for business owners who plan on raising capital through outside investment. LLCs are not public structures and do not have shareholders, so taking a company public is not an option either. However, in the event that you'd like to take your business public you may switch to a public legal structure, like a C corporation, later on.
On the other hand, if you are not responsible for its tax, then you should declare the equipment in Part III of the Business Property Statement (Equipment belonging to others). Where equipment is declared in Part III of the Business Property Statement, the Assessor will also send a Notice to File to the person reported as the equipment's actual owner.