Name Change

Not sure where to post this. Recently got married to someone from out of country and my spouse is looking to change her middle name. I’ve been on call waiting with the social security office forever and won’t be able to call them during my work hours. We are not sure what proper steps or procedures need to be taken. Does anyone have a solution or know what the steps are to changing a middle name? Will we have to wait for her green card to finalize before when can change her middle name?