Disclaimer: Not mine.
The thing is, she likes her name. It's a good name and after twenty-nine years, she's pretty used to it. What's more, she's proud of it. The secret thrill she gets from informing her future in-laws that she's keeping hers in favor of theirs--well, that's just a bonus. Her own little act of rebellion, and just another reason in the ever-growing list for them to find her terribly unsuitable.
Logan laughs when she tells him, and jokes that maybe he'll take her name. He's figures he's already left the family business, so why not ditch the name while he's at it? Logan Gilmore. It has a nice ring to it. Kind of. Okay, maybe not, but it's a nice thought, and in the end, he doesn't really care what her last name is as long as he is Logan and she is Rory and they are together.