Without a last name, this search will be next to impossible.
However, your best bet is to continue to press your mother's family. Ask about things like where your mother lived and worked when it is most likely for her to have conceived you. If that's a small town, you might have a chance. Also ask about anything at all she may have said. Her saying, "I miss Joe
." or "Can't let Susan, his wife, find out about us
." those are clues, but again they are helpful but not dispositive.
But it is slow going, and success is far from guaranteed. Best of luck to you.