You file an amendment with the state where the LLC was formed. It is the same company with a new name: you do not form a new LLC, and the IRS says you do not need a new EIN just because you changed the name.
Each state has its own form and fee. Two examples we checked against the states' own documents: Florida uses Articles of Amendment to Articles of Organization, and the filing fee is $25. Texas uses Form 424, the Certificate of Amendment, and the fee is $150, filed online through SOSPortal or by mail to P.O. Box 13697, Austin, TX 78711-3697.
The new name has to follow the same rules as a new formation. Florida's form says it must be distinguishable from other names on the state's records and must end with "Limited Liability Company," "L.L.C." or "LLC." Florida also tells you directly that it no longer does preliminary name searches or reservations, so search the state's database yourself before filing.
Check who has to approve it. Texas's instructions point to section 101.356(d) of its Business Organizations Code, which says an amendment to an LLC's certificate of formation needs the affirmative vote of all of the company's members. With co-owners, get that approval in writing before you file.
Then tell the IRS, which keeps its own record of your business name. The IRS business name change page sets out the steps by business type: a partnership, which is how a multi-member LLC is normally taxed, marks the name change box on its Form 1065 (page 1, line G, box 3) or writes to the address where it files if that year's return is already in. A single-member LLC owned by one person follows the sole proprietor route: a signed letter to the address where you file your return.
After that comes the list the state and the IRS do not handle for you: the bank account, payment processors, business licenses and permits, contracts, invoices, your website, and any DBA registered under the old name. If all you want is to trade under a new brand while keeping the legal name, a DBA may be all you need, and it is a smaller filing than an amendment.