We'll fix these by lashing and whipping the arms. This is a very secure and durable fix. And it looks good, to boot!
Step 1: Failed Fixes
The next thing I tried was superglue + heat shrink tubing. That worked ok at best. The superglued section broke and then just the tubing kept the arms on. Due to the tapering of the arms, this at least kept them connected. But they looked horrible, because the arms were bent at an awkward angle at the break.
After a couple years of that, I got tired of the glasses slipping off my face because the broken arms couldn't squeeze my head. I nearly lost them overboard a few times while sailing.
So, I set off again to fix my beloved cheap sunglasses. I knew I needed to brace the broken arms and to hold them more tightly than the heat shrink tubing could. So, I used a straightened paperclip and black duct tape! I was on the right track, but ...
It would have worked ok for a few months, had I flattened the paperclip, first. But, really, duct tape, while cool, is too shiny and my glasses looked even more ghetto than they did with the heat shrink tubing.
I needed a real solution.
Step 2: Lashing + Whipping
Next, I took my cut paperclips and hammered them flat on an anvil. I used bigger, thicker clips for this, not the normal small diameter ones. Be careful not to make them too flat, or they won't be rigid enough. When right, they are very nicely firm in every direction. Next, lash the paperclip to the glasses arms using what's called Common Whipping. Essentially, we tightly coil a stout (but thin) cord around the arms and clip, and together they securely hold everything together. The end result is both handsome and much stronger than anything previously used. The cord here is #6 tarred nylon--what Dave Canterbury calls "bank line". The Common Whipping is very easy to learn and it nicely hides the cord ends. It also allows one to really cinch up the coils--by pulling the ends.
I hope you find this useful, and please let me know if you have any questions.