How can I tell which wire is positive on this little repair job?

On cheap electronics red is almost always the positive lead, so I'd start there. That said, it's a convention, not a law, and no-name boards sometimes ignore it, so it's smart that you're double checking before you commit iron to it. A few things to hunt for before you solder anything. Look really closely at the board near those two pads, even with a phone flashlight and magnifier. Sometimes there's a tiny + or - printed in the copper layer, or a small triangle symbol pointing at the positive side. Also check the rest of the board, not just the broken pads. Electrolytic capacitors, the little cylinders, have a stripe down one side marking the negative leg. If you can see which side of the cap connects by trace to one of your two pads, that pad is likely negative. Diodes and the LED chips themselves have polarity too, so you can sometimes follow the copper lines with your eyes.

Another angle: figure out where the other end of those wires goes. If they run to a plug-in power adapter, the label on the brick will show a polarity symbol, usually a circle with a dot in the middle and a plus on one side. That tells you the center pin polarity, and you can work backward from there. If you've got a multimeter, even a $15 one from the auto parts store, that's the clean answer. Set it to DC volts, plug the light in, and touch the probes to the two pads. If the display shows a positive number, your red probe is on positive. If it reads negative, swap them around. No guessing needed. One more tip for the repair itself. Since the pad tore off, scrape a little of the green solder mask off the copper trace a few millimeters back from where the pad used to be, tin it, and lay your wire along that spot instead. Trying to solder to bare fiberglass never sticks. A bit of electrical tape over the joint afterward keeps it from pulling loose again the next time somebody brushes against it.

I'm actually working on a tiny timer board and running into a similar issue where there are no labels or minus signs to help. It's making me nervous about reversing it.

Since I can't find a polarity mark on the board itself, I guess my only bet is tracking the power from the wall jack connection. If I can identify the ground point that connects to the adapter, it should force the polarity of the whole setup. Have you ever tried tracing the power line backwards from the plug into the circuit to figure out the starting point?

I once worked on an old power supply board where the ground trace was actually highlighted in green on the silkscreen to make it easier to spot during repair. If you can find a trace that runs to a large ground symbol or a component that is definately the negative side, you can often trace back to your red and black wires. When I had to repair a similar fixture, I used a needle probe on the multimeter to check for continuity between the two pads and the known ground point. That little difference of just 0.1 ohm gave me the answer I needed before I even touched the iron.

That advice about looking closely at the trace connections is definitely solid. It is always a relief when someone points out the physical possibilities because guessing blindly is just asking for trouble with the power supply. I once had to fix a similar small driver board where the polarity wasn't marked, but the LED itself had tiny etched markings right on the casing. If you can identify the main chip or the LED that the wires go to, you can often see the polarity indicator lines printed right onto the component body. That gave me the definite answer without needing a multimeter, even though tracking the circuit is a good fallback.

Update: went with the red wire idea and traced things out first. Finding where the trace runs to the big component acutally settled it for me, so I soldered it back on and the light works again. One thing though, the board looks a bit scorched around my repair. Is that normal from a pencil iron, or did I cook something?