Do first names count for Social Security purposes?

Q: An employee’s Social Security card lists her first name as “Michelle.” She says this is a mistake, and her name is really spelled “Mechelle.” When I told her that she needed to get a replacement Social Security card, she said that the Social Security Administration told her that she would need to first have her name changed in court before a replacement card could be issued. This sounds fishy to us. Does the SSA care about first names?