N.F.
It didn't cost me a penny to have my name changed on my SSC. All you need is proper paperwork and identification and to fill out a form. You should be able to download a form from the website http://www.ssa.gov and also have any questions you have answered.
As far as importance it is important because you have a legal document stating you are married and have chosen to take his last name. What is on your marriage certificate should reflect on other identification documents.