I had found my father first, the few clues that I had lead me to believe that I was given his surname. My father refused to tell me her first name even though through records I had found her "maiden" name~lol, he was scared that he owed back child support...derp.
If your friend's mother's surname is common, it will take longer. But use the clues that you have and send private messages to people whom you feel could be a lead...or even a family member. I had re-read the non-identifying info I had thousands of times, like a detective trying to put pieces together that told a story. I was able with just two short paragraphs to describe who I was looking for along with a short plea for assistance.
It is an emotionally taxing journey. Having to spam strangers for help is trying. Hang in there, you are being a very good friend to assist someone as they travel this path.
I was really coming close to finding my mother when I finally got the courage to ask a Search Angel to help me. Within one day my Angel found my mother. Here's the thing, a real search angel is absolutely free~no strings attached. Angels usually are adopted people or mothers of relinquishment who do this work because it is right and want absolutely nothing in return.
Finding family for adopted people is difficult because of the secrecy and altered legal documents. Take advantage of every opportunity, but there are lots of opportunists who would like to take advantage of people with little or no recourse. Anyone who wants money to help your friend search has no better resources than those angels who will do it for free.
Feel free to message me if you like. Best wishes.