It's very simple, an article I read said it best:
"get your hands on that license!"
Once you (or your brother in this case) get your notarized copy of the license, take it to the Social Security Administration. You can do this by mail but every newly married bride I've spoken to said going in person is best because you have to also give them proof of who you are, like your driver's license, and it has to be an original, not a copy. You don't really want to mail away your DL. So, take the marriage license and driver's license and go to the SS administration office. You'll get your new SS card in a few weeks, then go to the Drivers License bureau, sign a new W2 at work and have work HR get your new medical cards ordered, and then call credit cards, cell phone, magazine subscriptions, etc. - many of these let you change your name on-line too.
No one asked for proof except SS and DL and work kept a copy of the license on file.
Hope this is helpful!