While I m not licensed in the State of Ohio, I will try to give you some guidance here. Initially, there is no set formula or chart which sets forth when a father can have overnight visitation. Cases such as your are extremely fact sensitive. No two cases are exactly alike. The Court will take into consideration numerous factors, including: the ages of the parties; whether he has made suitable overnight sleeping arrangements for the child; whether there will be other friends or family members present who might assist him; the need for the father and child to bond; whether he has previously exercised his right to non-overnight visitation, and done so responsibly; the cleanliness of the environment. There are numerous other facors as well.
The Court's primary concern will be what is in the best interests of the child. If you feel that your child's father is otherwise responsible, but are concerned that the chid is too young, you may want to give it a shot on a limited trial basis until you feel more comfortable with the overnight visitation. Your motherly instincts will tell you right away whether it is working or not.
There is no "age". All he needs to show is that he is capable of handling the child for an extended period of time in that overnight parenting time will not jeopardize the child's wellbeing. If he can show he is able to change diapers, feed the child and properly provide care, he can have overnights.
The big restrictive factor is feeding - namely breastfeeding. If the child is exclusive to the breast, then overnights become problems. If the child is being bottle fed, then he can do this and therefore overnights are possible.