In general, I would check for shorts or bad solder joints. There’s surprisingly many points where things can go wrong, even with such a small circuit :)
That said, you say you’ve got a 9V power supply (measured?) and are measuring around 8V on the output, so at least it seems unlikely that you have a short circuit from your power supply to the output directly, which would be difficult to do with the board laid out as it is anyway.
There’s basically a few sections to the circuit (ignoring the regulator itself), input, feedback and output.
If you look at the schematic, you’ll see that your power supply input goes through a diode (with a small voltage drop), and a PTC resettable fuse (with a negligible drop unless it’s heating up) before it gets to the regulator. You’ll also see a small test point named ‘RAW’ there. There’s no associated pad on the board, but you should be able to measure at the inside pin of the PTC (right next to the on/off switch). If this voltage is the same as the voltage you’re getting at the output, it seems likely that you have a short between the input and output pins on the board, most likely at the regulator itself.
If you see a higher value there, then the regulator is at least doing something, but not what it’s supposed to. One possibility there is that it doesn’t have the appropriate feedback on the adjust pin – this is the pin that leads up to the 3 resistors in the middle of the board. Check the solder joints at those resistors, perhaps one of them didn’t quite get soldered down right making the regulator try to output a very high voltage, getting limited by your power supply, the voltage drop in the input section, and its own drop. Normally the combination of the 390 Ohm / 240 Ohm, or (390 Ohm + 330 Ohm) / 240 Ohm, are what set the voltage (nearly 3.3V, and 5V respectively).
Lastly there’s the output section, but there’s nothing particularly exciting going on there. I’d be cautious leaving it powered in its current state for long, though, as that LED can’t be happy about the excess of power it’s getting :)
The above are just guesses, so good luck, and please do follow up if you find the culprit :)