When the screws holding the hinge to the screen in my laptop gave up the ghost, I needed a quick fix. A little tack acting as a rivet does the trick.
Step 1: Remove Bezel
I won't go into too much detail on how to remove the bezel. Every laptop is a bit diffrent, so round up the tools appropriate for yours. I will say that an old knife with the tip ground off makes a great tool for opening up plastic casings.
In the first photo you can see the broken hinge, normally that metal plate is captive between an inner plastic frame and the bezel. There is a lot of torque on the hinge when opening it and when the screw failed it resulted in the bezel breaking.
Step 2: Hammer Time!
Drill 1/8" holes through the hinge and the back of the laptop. I reused the holes that the screws came out of, and left the plastic frame behing the hinge as well. The frame maintains the alignment of the screen, so even if it's not doing much for the hinge, I don't want the screen moving around.
Next insert the tacks from the back of the screen through the holes you just made. It is a tight fit, so use a nut driver or socket as a hollow punch to drive the hinge down over the tack.
Once the tack is flush with the back of the screen, use a punch to fold the tip over or peen it to lock everything in place.
Step 3: Revel in the Renewed Hinge!
Pop everything back together (reinstall bezel) and you're set!
I think the fix came out nicely. Adds a bit of contrast to the back of the machine if you're into that sort of thing, and works just fine regardless of aesthetics.