Is there an easy trick you can do to remove a broken part in a hole?

I don't have this model, I've only looked at photos online, but I assume the broken part is a small white metal casting with a stem on the back which is glued into a hole in the body, and that stem has snapped off. If you have a very, VERY fine drill bit it might just be possible to drill into the broken stem then pull it out, but being very soft metal it would probably be wrecked in the process.

You could take the model apart and see if the hole they drilled in the body to take the stem goes right through the body, in which case you might be able to push out the broken piece from the inside (unless it is glued in too firmly). 

Otherwise I think it would be almost impossible to remove, and the best answer might be to drill it out completely. Then superglue a new pin (a small piece of plastic sprue or a small piece of metal rod of a suitable diameter) to the back of the lamp/licence plate unit and glue it into the hole in the body. It won't be very strong, but then you are not going to be playing with it, are you?

Maybe someone who has this model or has repaired a similar Brooklin part  will have a better idea!

Michel, I don't have this model but I just looked at pictures of it.  There are three for sale on eBay, and all of them show the Buick with the tag/stop light as a single fixture for buyer attachment.  Not knowing what glue that buyer used in his attempt, I suggest attempting to dislodge the broken insert with a fine needle.  That probably won't work unless a simple glue was used.  Next you might try a glue dissolving agent, such as Super Solvent for cyanoacrylate.  That might loosen the glue enough for the needle insert to jiggle the peg loose.  If you have the major piece, you might try to affix it to the broken peg with a very small touch of a fast-acting glue.  And when none of these works, ask Amanda for a replacement piece and then drill out that piece as Graeme suggests.    Good luck!  

If this is a Brooklin I have found that, at least on older Brooklins, these type parts were attached with white  glue, similiar to Elmers wood glue.  In fact Brooklin recommends  buyers to fit them with white glue as it has no effect on surrounding paint.  To remove simply soak that area of the model in a bowl or glass of warm water and it will loosen and can be pulled or pushed out.  The good news is that Brooklin offers a GREAT service in replacement parts.   You could also create a replacement with a small section of paperclip wire attached to the license plate with a very small dab of J B Weld.   I've created many replacement rearview mirrors this way.