The mother should contact a lawyer who can contact the father and ask him to sign a release permitting the child to be adopted. If this fails, the mother should petition a court to terminate the father's rights as a parent. This can be done along with the adoption proceeding and need not be a separate action. The mother can establish the father's abandonment by her testimony and by presenting evidence that the father has failed to provide any support or visit the child. Remember, always try to get the consent by non-judicial means first.
Persuasion works better than a confrontation in court.
Choose a lawyer who will work with you on this and the level of stress will be greatly reduced.