In the Syrian and Indian Churches, godparents take the child to be baptised at a certain point in the baptismal service from his/her parents and make the baptismal vows in his/her name. They renounce Satan and his evil, and accept Christ and profess the Orthodox faith. This is done by repeating the vows as recited by the bishop/priest. To perform this service to the family is considered a great honour and privilege. After this is done, the godparents are responsible for the child in a special way, and especially for his spiritual upbringing.
That's the theoretical view, anyway. In Indian Orthodox practice, this has largely become a family matter. The godfather of the first son born to a man is the child's paternal grandfather, the godmother of the first daughter is the paternal grandmother, the second son, the maternal grandfather, and so on (boys get godfathers, girls get godmothers). Choosing someone outside of the family for this is extremely rare in my experience. Because this has become a family thing, the godparents' responsibility is generally not emphasised (it is assumed that if you are family you will do all these things anyway).
I am not sure what the Syriac Orthodox practice is in regard to this (although, based on our Indian experience, I have a hunch); it would be best for you to ask these questions to a Syriac priest, deacon, or layperson to see how the duties of godparents are exercised.
You say that in the Coptic tradition there really isn't a place for godparents. May I ask what the Coptic baptismal ceremony is like, and specifically if there are any vows/renunciations, and what happens? Is the service online somewhere for us to read?