Since my wife just went through this I will post our experience.
1. to move quickly after US entry we applied for a Marriage License in Colorado before going to the Social Security office. The County required her to sign an affidavit stating that she did not have a SSN at the time we applied for the license.
2. We went to the Social Security office with all the required documents (her Argentine passport with K1 Visa and POE stamp, I-94 form and just in case, her I-797 form from the I-129F submission). All was OK and they approved the SSN request using her maiden name.
3. We got married, submitted the completed marriage license and waited for the recorded Marriage Certificate to come back in the mail.
4. We went back to the Social Security office with the Marriage Certificate and requested her name change due to marriage. No problem at all and we got the updated SS card in the mail days later. That was the first official name change document.
5. I completed and submitted the I-485, I-864 and I-131 using her new married name which should result in an AP and green card in her new name.