I have now seen STXI four times!! <--- 
And still I can't see where that little metal model of the Kelvin that Kirk turns over in his hand after Pike leaves him comes from. Does Pike drop it on the table in front of him or.. (and I think this is the best answer) does Kirk take it out of his pocket? Taking it from his pocket would be a wonderful gesture because it means he carries a small toy model of his father's ship around with him, a past he never lets go of despite being so flippant with Pike about it. The model even appears to be broken, missing a nacelle, so we can assume this is something Kirk has had since childhood. Very poignant.

And still I can't see where that little metal model of the Kelvin that Kirk turns over in his hand after Pike leaves him comes from. Does Pike drop it on the table in front of him or.. (and I think this is the best answer) does Kirk take it out of his pocket? Taking it from his pocket would be a wonderful gesture because it means he carries a small toy model of his father's ship around with him, a past he never lets go of despite being so flippant with Pike about it. The model even appears to be broken, missing a nacelle, so we can assume this is something Kirk has had since childhood. Very poignant.