Unregulated soldering irons can be dialed down using a variac or a speed control box. You will find the temperature needs to be just a bit more than the minimum needed to melt the scrap welding rod material, because the base material acts as a heat sink. Rig a fan to one side at a distance to gently blow fumes away from you, otherwise the body heat from you makes kind of a chimney and the smoke and chemicals come straight towards your face!
If you just dropped melted material to the surface it would solidify too quickly and not bond to the base. If your iron is just a bit cool you can melt both the rod and the base but the mush cools too quick to mix and bond well. Turning up the heat a bit forms a shallow liquid puddle on the base material and you can melt the rod material into the puddle by pressing it on the tip where the tip meets the puddle surface. Too much heat and you melt thru the base, and also make a lot of unhealthy smoke from boiling plasticizer and possibly vaporizing and oxidizing plastic material. The plasticizer is needed to keep the plastic from getting brittle so you do not want to boil much of it away.
Just like welding metal, a thick weld is built up in successive passes where in each pass a relatively thin layer is laid down. The relative thinness of each layer facilitates heat penetration depth control. Because plastic conducts heat poorly there is plenty of heat available and a temptation to brute force lay down a lot at at time. But this risks both melting thru to the far side, and poor bonding of the deposited glob except at the point where the iron is touching the base. With the temperature just right and a light touch, you can melt some rod material onto the base and then melt it into the surface to a controlled depth to form a good bond with the base by wiping the iron back and forth along the surface.
The back and forth motion of the heating source is needed for most welds of moderate size. If a soldering iron is used it must be held at controlled depth as you move it back and forth...otherwise it has a tendency to go with gravity, sink in and melt thru to the far side, a bad thing!
Threading a self tapping screw into a drilled hole puts tremendous radial splitting and rotationally shearing stresses on a boss. Way to much for aged plastics. You may wish to experiment with pre-casting a boss around a screw (coating a screw with mold release material like a thin lacquer then building a boss around it using a welding technique), backing out the screw (to make sure you can) then welding the internally pre threaded boss to the case. You might want to use a hinge as a temporary alignment fixture to hold the boss in exact position then tack weld the boss in place before removing the fixture and doing the complete weld build up. Be sure the screw is inserted deep enough during the casting phase to mount the hinge with a bit to spare. If the screw bottoms out during final assembly, tightening it further will wreck all your previous weld work.
Just like metal welding, beveling the boss base would allow you to get good bond to near the center of the boss base, and then lay down circular rings of weld material working outwards to eventually form the volcano shape sloping wide base. Best to overdo the support of the boss, then use a Dremel to cut away any clearances needed for hinges and LCD assembly. In any case the Dremel (or some files and scrapers) is needed both to clear copper shielding metal from the weld site and to smooth the final welded surface to something presentable. [

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Another repair technique to consider (I prefer welding but this could work too) if you have access to adhesive bondable plastic raw material and carving tools: use a machining or sculpting techique to shape rectangular block hinge mounts (cutting out clearances for the hinges and hinge springs) with very large base area matching the inner curve of the case (after you cut all the copper and reinforcing ridges and broken boss stubs away). The idea would be to have blocks about 1 inch deep by 1.5 wide just thick enough to hold the hinges at the right height above the case. These blocks could be glued in place with a superglue rated for plastics including the ABS and PC used in cases. Adhesives are normally much weaker than welds, but this can be offset with sufficient bond surface area.