How about making flair look just like our representation on SO and related sites.
I agree totally that it should be more distinctive about the site name.
And yes, the entire image should be clickable, not just the gravatar and id.
So how about something like this: (With apologies to Jon)
Edit: Okay, okay, so I'm not the best artist. One of you can come up with something better.
But the idea is this: The flair, the way it is now, does not say StackOverflow or indicate what it is. You hover over it, and if you are lucky and hover over the image or id, it might say "see my profile on StackOverflow".
You don't have to like my artwork, but I'd be surprised if noone agrees with the idea.
And did you -1 me because I used a .jpg???? I should flag that comment as hate speech (someone who hates jpegs).
Andrew Moore offers another design that at least has StackOverflow front and center in it: