I had exact same question before I filed my I-751. I didn’t find any better answers except filing I-90 after I get my unconditional GC, so I just took a chance… and I filed I-751 with my married name.
I included my marriage certificate as a proof of legal name change and I listed all names used in my application. So far, my first NOA, my FP notice and my duplicate NOA came under my married name, just like I wrote in the I-751 application, no questions asked, no RFE seen…
My application still in process, but I expect my new, unrestricted, GC be in my married name.
~~~~~~~~~~
Question to girls who replied before me – why are you talking abut court? She is not trying to change her first name or take on some makeup last name. She is changing her last name to her husband’s. Marriage certificate is the legal and sufficient enough document that lets you change your last name without going through a court on all your immigration papers. And form I-90 (Application to Replace Permanent Resident Card) is used for a last name change on GC. The only thing is - she cannot file I-90 before she gets unconditional card.