maybe try a Jeweller, i know when I had to fix a pin in my watch they were able to fix it with a brass pin, they might have a pin or rivet small enough and they should do a nice enough job putting it in, just a thought
+1 on this idea, but it just might be just as easy (and cheaper) to replace it.
While your friend means well, I don' t think a "pop" style rivet (as "gun" leads me to think) is your answer, nor would it look right.
Maybe craft your own pin, and a tool of sorts to buck it with?
I'm not sure of the size required, but you might be able to fashion something out of a silver colored alloy brazing rod of sorts, aluminum coat hanger, or a brass rod from the craft/hardware store.
Again, it may just as well be best to find a replacement.
Cheers!