Myra and I would like you and Elaine to be the godparents of Steven( [color=red]Or Steven’s?)
Thanks
Neither. Possession by people uses the Anglo-Saxon genitive: ‘Myra and I would like you to be Steven’s godparents.’
The expression I heard is “Myra and I would like you and Elaine to be the godparents of Steven.”
It is from Seinfeld. I know a lot of expressions used in the show are informal. Since it is used by native speakers, I guess it is not actually wrong.
With “a friend of mine” and such expressions, the choice among the three structures is becoming a real headache. 
It doesn’t sound natural.
I would use ‘to be Steven’s godparents’ too.
Yes, perhaps within the script we could see why it was phrased that way.
Jerry(to George): Will you shut up.
Elaine: I’m taking a cab home. I can’t take it anymore.
Jerry: So what were you about to say, Stan?
Stan: Myra and I would like you and Elaine to be the Godparents of Steven.
Elaine: Wow.
Jerry: Me? A godfather?
Just this, nothing much, so I guess it’s just a careless informal way to express the idea?
(1) As I understand it, either of the following is correct:
(a) I am a friend of Tom’s.
(b) I am a friend of Tom.
(2) Some people feel that there is a little difference of meaning between the two; “everyone” agrees that “Tom’s” is the most idiomatic and popular; and some people feel that “Tom” is – grammatically speaking – also “correct.”
Both are OK. The problem arises with this:
Here is a photograph of Tom.
Here is a photograph of Tom’s.