I believe (and I'm sure that the nurses and others in the forum will lovingly correct me if I'm wrong
), that if the mucous has a green color to it, that indicates an infection. I myself deal with discharge from my eye a lot.
From my own experience, I get more of a discharge in the cold, rather than in warm weather, but everyone is different.
In my opinion, it might be helpful if you could remove the eye and irrigate the socket. I understand that she doesn't want you to remove it and she is only 5, but perhaps perhaps if you got a stuffed animal and somehow was able to remove it's eye (a Mr. Potatohead toy might work) and showed her how you would "clean" the toy's socket first, that might help (or not, again, I don't have children, so it's just a thought).
When I take my eye out for cleaning once a week, I often irrigate my own socket with the saline solution that I use for the contact lens for my good eye. It's cold and annoying, but sometimes my socket gets irritated if I cry too much (which I've been doing lately) and it does make my socket feel better.
I hope this is helpful.