An excerpt from my catechumen class on Godparents is below:
The institution of sponsors (god-parents), who serve as witnesses and guarantors for the faith of the person being baptized and are obligated to edify him in the rules of Christian life, has existed since the first century of the Christian era. Church literature of the second suggests that the sponsors of the first centuries were usually deacons, deaconesses, hermits, virgins, and in general persons dedicated to the service of the Church and thus capable of edifying the newly-baptized in the truths of the Christian faith and its ethical principles. According to the "Apostolic Canons" (3, 16), a male Christian was obliged to take one deacon, and a woman one deaconess as sponsor. This practice has been maintained in the Church ever since, i.e., a person baptized is required to be sponsored by one person of the same sex. According to the Rudder (ch. 50, pt. 2), the person baptized, "when he leaves the saving bath, must be received by one faithful person.
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"The Responsibilities of the Godparent only begin at baptism, the role really expands and hopefully blossoms as the Godparent and Godchild develop a close and loving relationship. As with any relationship, this spiritual one needs to be fostered and cared for in order for it to develop. The best way for this relationship to grow is through prayer. Pray for your Godchild [ if a minor child pray also for his or her parents], and the parents should encourage their child to pray for the Godparents. Just a minor Godchildren pray for the Godparents likewise the adult Godchild should regularly pray for their Godparents. By doing this you are encouraging a relationship and giving it the spiritual basis on which to mature.
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An often overlooked area of responsibility is The Responsibility of the Godchild
Godparent and Godchild should develop a close and loving relationship. As with any relationship, this spiritual one needs to be fostered and cared for in order for it to develop. The best way for this relationship to grow is through prayer. Pray for your Godparent and his/her family. By doing this you are encouraging a relationship and giving it the spiritual basis on which to mature.
When greeting one's Godparent, you should feel the love and familiarity that you have with your own parents. It is NOT inappropriate to hug or kiss your godparents, as you would your own parents. A Godchild should light candles and pray for their Godparents every time they enter a church, say their family prayers, and say their personal prayers. The Godchild should observe the Godparents names day. Celebrate it with a special visit and dinner if you're nearby, and give a "spiritually oriented" gift to celebrate, like a spiritual book of the Godparent's patron saint's life, a new icon, etc.Keep in touch by phone, e-mail, or postcard if your Godparent lives out of state or across the globe. Prayer and love in Christ know no distance!
There will come a time in which your Godparents have aged and are less able to be fully present with you due to illness or perhaps a nursing home placement. Remember to continue to pray for them and visit or write them often to maintain your relationship. Ask for their advice even though you have grown up.
Finally there will come a day in which your Godparents will repose in the Lord, maintain your image of your Godparents in your mind to help brings peace and memories of love and wisdom. Pray for your Godparents and offer memorial services in their memory, do works and offer alms in their name. And pray for them as they will continue to do for you in heaven.