How do I go about finding my Mothers 1st & only son when I know only his birth name. Mother wants no contact

You don't. If she does not want contact, then most likely she is unwilling to give you anymore info than you already have.

Respect her wishes, she had her reasons for giving him up and she has her reasons for no contact. You must respect what your mother wants.

I think you have a right to know your siblings... is there a particular reason why you want to find them? If possible explain to your mom why it is important to you. If she still won't give anymore information it will be a hard search but it can be done.