Thought (b) is the right one. If there is no father in the appropriate column of the birth certificate it means that the mother was not married and the "putative father" (that's the technical term) was unwilling or unable to admit paternity, attend the registration and sign the register. If your grandmother's maiden name had been "Cook", then it would have said "formerly Cook", so it looks likely that this was her occupation.
Your parents' marriage certificate might possibly offer a clue, but I wouldn't hold your breath.
Even if the chances of finding the identity of your grandfather are slim, it is still possible to trace the family back through the female line. Did Eileen marry the father of her other children? Her marriage certificate will show her father's name and occupation.