Two Methods:
1. Center drill (tiny hole in this case) and turn out counter-clockwise with an easy-out inserted in a Tap-handle. Available any hardware store.
2. Use small hole saw (larger than the piece you want to extract) and refill & finish the hole. Although you could incorporate the hole with an alternative method of mounting the snap stud you will need in that area. The hole saw incorporates a center guide drill bit that may be difficult to use with the broken stud in the way. You could attempt it w/o the guide bit because it is so small. Either way, hole saw should turn counter-clockwise till a groove is established, in order to get through the gelcoat without chips. Once established, continue cutting in a clockwise manner.
I did see a new product at the Strictly Sail Show in Chicago, that I believe extracted screws with a "frinction Bit." but I don't have any information on it. Only remember thinking how handy it would be in the situation you asked about. Sorry. Just know there is something out there to help.