The short answer
For an online or service business owned from abroad, with no people or premises in the US: Wyoming or New Mexico. If you plan to raise venture capital: Delaware. If you want a large, recognisable state on your invoices: Florida or Texas.
Wyoming
Wyoming invented the LLC in 1977 and still competes for them. The filing fee is $100 (plus a transaction fee of $2.40 charged by the state), approval usually comes within one to three business days, and the annual report starts at $60. The owners are not listed in the public formation document. There is no state income tax. It is the default choice for most non-resident founders for a reason.
New Mexico
New Mexico is the cheapest state to keep an LLC in: a $50 filing fee (plus a $1.95 transaction fee charged by the state), and then nothing. No annual report, no annual fee. Owners are not listed publicly. The trade-offs are small: the name is less known than Delaware or Wyoming, and some banks and partners simply see fewer New Mexico companies.
Delaware
Delaware is where most US startups and listed companies are incorporated. Its business court and its body of company law are the reason investors ask for it. For a small foreign-owned LLC those advantages rarely matter, while the costs do: a $110 filing fee, a flat $300 franchise tax every year, and slower standard processing unless you pay to expedite.
Florida and Texas
Both are big economies with no state income tax on wages, and both names read well to customers. Florida charges $125 to form and $138.75 a year. Texas charges $300 to form; its franchise tax report is due yearly, though small companies fall under the no-tax-due threshold. Choose them when the state itself matters to your business, for example because your customers or partners are there.
What does not differ
Federal tax is the same in every state, and so are the IRS filings for a foreign-owned LLC. Liability protection is comparable. And if you later hire staff or open premises in another state, you register there regardless of where the LLC was formed.
This guide is general information, not legal or tax advice. Rules and fees change; check your own situation with a professional.