The role Connor plays in this episode is very clear and obvious. This episode basically comes to show how much meaning this child gives to Angel's life, the real value he has in his father's eyes, and that it's priceless. In a point, Angel realizes he doesn't really need anything else, that nothing is really as important as his son to him.
Throughout this episode, Angel feels left alone, outcast and isn't important, he feels he's lost his place among his friends, that they no longer care about him the way they used to, and most importantly - that he can clearly (in his eyes) be replaced - with Groo. The way he sees it, Groo appears to take Cordelia from Angel, to take his mission, and his position in the group. But it all, though still hurts, means as nothing when he realizes what he does have; when after he returns from the night at the ballet and Cordelia leaves him to go with Groo and he feels his heart is broken. But it's suddenly not as broken when he simply sits for hours by his son crib and watches him, watches him stir, watches him smile…his mind begins to form different perspective on things that really matter.
And in the end, after he gives Cordelia money and sends her away on vocation with Groo, he picks Connor up and hugs him, and when he sees his son happy and smiling, he realizes he's really not alone, that the real meaning to his life is right there in his arms, and will always be with him.
Angel: Working late?
Wes: Yes. You startled me.
Angel: Oh, we didn't mean to.
Wes: I thought I was alone.
Angel: Yeah. So did I.