Harry Potter (the child’s godfather) and Andromeda (the child’s grandmother) becomes the surrogate parents because of the death of both the child’s biological parents (Tonks and Lupin). During the final battle in “Harry Potter and the Deathly Hallows”, Remus Lupin leaves his wife and child to go and fight. Tonks cannot bear to be away from the action, so she also goes and fights. During the course of the battle, Dolohov kills Lupin and Bellatrix kills Tonks, leaving their child, Ted, orphaned. The child is then raised by his grandmother, Andromeda, and his godfather, Harry Potter. The text says that Harry Potter is involved in raising the child because he knows what it is like to be an orphan. Andromeda would naturally be involved in the child’s life because she is the grandmother. When Harry was orphaned, he also did not go and live with his godfather. I do not believe that the “godfather” status makes a person the only choice for adoption when a child is orphaned.
Harry does in fact play an integral part in the boys life. In the very last chapter of the book, "Nineteen Years Later", there is a reference made to how often Teddy is over at Harry and Ginny's house in a week, and how he practically lives there.
Lupin and Tonks have Teddy when Harry is still just a child himself. Not only is Harry way too young to take care of him, and he has no experience with babies/children of any kind, Harry also has a lot to sort out after Voldemort is gone. He is distraught and needs to put Hogwarts and his life back together, which takes a lot of time. By the time Harry is finally put back together, Teddy is about to start school and comfortable with his current living arrangements.
At the time of the battle at Hogwarts, when Lupin and Tonks die, Harry is only seventeen years old. Even in the Wizarding world, seventeen is too young to take on the sole responsibility of raising a child. But, I am sure, based on the connection that Harry had with his own Godfather, Sirius, that Harry will play an integral part in Teddy's life.