I am writing this on behalf of my dad, I am quicker on the computer.
My dad has his original birth certificated with his birth mums name, how do we trace her?

Where was she from ( OK this is an international site, what country is she from)
Is her last name, common or less common

You try looking her up on a few of the dozen or more people searches.
You if her name is uncommon, try looking up people with that last name in the area where she used to live.

your hire a PI trained to do these things