WE LIVE IN USA
Your married name is decided the moment you apply for the license: miss it and you're going to court
The cost of changing your name after marrying in the US varies enormously, and what determines that gap is a very short window: the ten minutes the two of you spend at the county clerk's counter filling in the marriage license application.
That one box on the form
There's a field on the marriage license application asking what name you'll use after the wedding. Fill it in right there, and the certified copy you receive carries you through everything that follows: Social Security card first, then driver's license, then passport. Nobody along that path will ask you "on what basis are you changing your name," because the certificate itself is the legal basis.
Leave it blank and decide later, and in most states the answer is: go to court and petition for a name change order. That's a separate process: a filing, a fee, and in some states a published notice, with a wait measured in months. Same outcome, but one route is free and folded into the license fee, and the other is attorney fees plus court costs plus several months.
Why so many people leave it blank
People actually know they can change it, but their minds are just elsewhere at that moment. On license day, both of your attention is mostly on documents and checking if the date works out, so the name field looks like a "settle this later" spot. Plus, plenty of people haven't decided whether to take their husband's surname yet, so it gets turned in blank.
The safer bet: as long as there is any chance you might change your name, fill it in first. Filling it in doesn't commit you to using it. Although the certificate lists the new name, if you don't update your SSN or driver's license, in practice you are still your original name. The reverse is where it gets tricky: if you leave it blank, that shortcut is gone.
What you're allowed to change it to
The permitted combinations vary by state, but the common ones are: take your spouse's surname; move your original surname to your middle name and add your spouse's; or combine both surnames (some states allow a hyphen, some don't). Note that inventing an entirely new surname unrelated to either of yours is usually outside what a marriage license can do. That still means court.
Also, this field is for both spouses, not just the bride. A husband changing his surname, or both of you changing, goes in the same place.
The order afterwards can't be shuffled
Once you have the certified copy, the first stop is the Social Security Administration, not the DMV, not the passport agency. Both of those check back against the SSA database, and if the SSA hasn't updated, your DMV trip is wasted. After you file with the SSA, the federal database needs roughly a day or two to sync; go too early and you'll be turned away just the same.
One more thing that bears directly on the honeymoon: the name on your plane ticket has to match the passport you'll actually be travelling on. After a name change, a new passport takes six to twelve weeks, and honeymoon flights are usually booked long before that. So always book under the name currently in your passport, never the new one. It's the single detail on this list most likely to wreck a honeymoon.
In one line
The easiest chance you'll ever get to change your name is those ten minutes at the counter. If you're not sure, fill it in.