Parenthood I think begins a whole other level of interaction with the gods, and sometimes with gods that you would have rarely thought of before. For instance when you desire to have children you may find yourself praying to gods such as Aphrodite, Leto, Hera and Zeus, Pan and so on to deities concerning fertility and generation. When you are expecting you will find that your prayers may drift to gods concerned with childbirth such as Hera, Artemis and Eilythia. And then after the child is born you prayers will be often directed to gods who are concerned with the welfare of the child and his or her development.
This later phase of prayerful address seems to be one that lasts the longest until said child makes way into adulthood. Typically I think it could be agreed that the deities perhaps the best to pray to for the welfare of the children is that of Leto (the mother), Artemis and Apollon. Artemis and Apollon are particularly connected to the welfare of the young and their successful survival of not only the seven days following their birth but into the succeeding years into adulthood and the stages that they pass through therein. This triad seems to be the most important in the successful rearing of children, or at least is a fundamental core deities which I think most could agree about.
This is not say that other deities would not be of value and interest in protecting your child. I think those gods which are predominantly worshiped in your household would take an interest in the welfare of your child and would be beneficial to pray to as the continuation of your oikos. For the case of my soon to be born baby daughter, Amber Artemisia Rose, this means that I particularly would be inclined to pray to Apollon, Artemis, Leto, Zeus, Hera and Aphrodite as the principle core deities of my oikos. That is not say that the other Olympians and other gods are not honored and that there are not other important gods, but these are the core gods that I feel interact the most in the welfare of my oikos, and who have shrines established for that reason.
That aside, there is also the personal gods of the child’s day of birth that are in my opinion take vested interest in children born during their sacred months. This is according to the Orphic calendar which assigns the zodiac months to the gods in sequence, therefore in a given zodiac month there is the god/dess and his/her Orphic pair. I take this a bit further to include not only the sun sign but also the rising sign which is rising directly opposite of the sun. For myself, as Scorpio, this means I was born under the zodiac month of Ares, with Aphrodite as his Orphic pair (Aphrodite of whom has shown a great deal of interest in my life over the years), with my rising sign (being Gemini) being ruled by Apollon with Artemis as his Orphic pair. For my eldest daughter I have often prayed to Athena who rules the sign Ares, and her Orphic pair Hermes. For Amber Artemisia I will be directly prayers particularly to Poseidon for the month of Pisces and his Orphic pair Demeter.
I think that this is especially valuable during birthdays when the gods who have protected the child over the year be honored by the parents with offerings for the next year of the child’s life. It goes without saying that a parent could of course go further by placing images of the gods near the beds of their child, or even give them charms to wear that represent the gods. My eldest daughter for years had a minishrine to Artemis in her bedroom when she was very small for instance.