Re: '87 Mercury Blackmax 135Hp Temperature Sensor
If yours is like mine you have a thermal spst switch that is mounted in the water jacket(s) and held in place with a 1/4" SS bolt. Corrosion apparently took it's toll and you attempted to unscrew the bolt and broke it off. Preceding is conjecture, but if true, continue: If you can flatten the top of what's left of the stud, do it to give you a flat surface; works best. Using a punch, the brass spring loaded things from Harbor Freight do a great job, locate the center of the stud and operate the punch a dozen times to get a good hole for starting the drill.
Using a drill bit that is the size of the bolt less the threads, drill down into the stud half an inch or until you stop getting SS debris. Using an easyout of suitable size and good penetrating oil like PB Blaster, and a hammer to "impact" the stud, work the stud out.
Once out, get a tap of the same thread and size as the screw and clean up the threads. Reinstall the new sensor using Blue Locktite thread sealant which will help you to install the new SS bolt and help to remove it if needed in the future.
Mark