Taking a spouse's family name is ordinary. Doing it midway through an immigration case is where it gets expensive.
The problem is not the name itself. It is that USCIS, the State Department, your home country's passport authority and your employer all hold a record of you, and immigration works on those records matching.
The mismatch that causes trouble
USCIS will generally recognise a name change based on a marriage certificate. Your home country's passport may not change at all unless you apply to your own consulate, and some countries will not change it on the basis of a foreign marriage.
So you can end up with a green card in your married name and a passport in your maiden name. At the airport that is a question. At naturalization it is a discrepancy to explain, with documents, years later.
The order that usually works best
Update the passport first, through your own consulate, then use the new passport as the evidence when you file with USCIS. That way the two most important documents agree from the start and everything downstream follows.
If a case is already pending and the name changes midway, tell USCIS rather than waiting for the interview. An unexplained new surname on a document is the kind of small thing that turns into a request for evidence.
The simplest route, if you are close to citizenship
You can ask for a legal name change as part of naturalization, and the court grants it at the oath ceremony. The new name then appears on the naturalization certificate, and you use that certificate to update everything else.
If you are within a year or so of being eligible, waiting is often cleaner than changing the name twice.
What to keep consistent
Use one spelling everywhere, including middle names and accents. If your documents disagree on whether you have a middle name, or a hyphen, fix that too. Small inconsistencies are the ones that follow people for years.
Bring your marriage certificate, both passports and any prior USCIS notices to the consultation and we will map out which document to change first in your particular situation.