Connor didn't have any specific role in that episode, plot wise. He was just there, playing Angel's baby son, which is why I don't really know what to say at this part regarding him/his character.
He made very little appearances in it and with no particular meaning or reason. So though Connor did have a few screen minutes in that episode, except for an addition of inevitable Connor-cuteness, it had nothing in particularly to do with him.
Cordelia: Oh...what a cruel dilemma: presents of sweet little baby face.
Take the baby.
Angel: You're choosing birthday gifts over my kid?
Cordy: Take the baby! Take the baby!