Frequently, you'll run across arcade boardsets which are in untested condition; you see them being sold on the newsgroup or pick them up from your local amusement operator. These can be trash or treasure once you test them, but there's a few things you'll want to do before you get that far.
Visual inspection of the board is very important. There can be a lot of tell-tale signs that can save you a lot of debugging time down the road if you look carefully.
Depending on the condition of the board, you may want to clean it first. If it's been sitting on a shelf for years, it probably needs a good cleaning. If it's relatively clean, skip this part.
Let's start on the solder side first.
Check the bottom of the board for previous work. Are there surface wires or signs of resoldered devices? Check the work, and try and speculate why it was done.
Next, check for scratches, cuts and scrapes on the bottom. When boards get stacked or boxed together, devices and heat sinks can scratch the bottom of boards and cut traces. If you see any deep scratches, use a multi-meter to check continuity along the traces to make sure none were cut. I bought a few boxes of boards from Great Western Trading company a few years back, and 50% were fixed by simply repairing cut traces on the solder side. Sometimes a scratch will smear two traces together, so you should check them for shorts, too.
Also, sometimes the IC legs stick so far out the bottom that they can be folded over onto other traces. Make sure the IC legs are not folded over and shorting to another trace it's not supposed to. Check for solder blobs or splashes that may be inadvertently shorting two lines together, especially if there is evidence of previous amature work.
Resolder any and all header pins; these are the pins which stick out perpendicular to the boardset where the connectors attach to. Williams boards, along with Tempest and Battlezone boards, are notorious for having cracked solder joints at the header pins.
Now, flip the board over to the parts side.
Check socketed devices for corrosion; I've seen IC legs that look fine in the socket, but crumble to dust when removed. I've been told to use an eraser tip to clean the IC legs, but for years I've used a small piece of sand paper and just sanded LIGHTLY and haven't had any problems. If the socketed devices look clean, it doesn't hurt to reseat them; pry them up with a small screw driver and push them back in. Check to make sure that the IC legs do not fold when you re-insert them. Verify that all devices are correctly oriented (all their "notches" face the right way).
Also, check for missing devices. Sometimes operators will pull valuable chips from boardsets. However, not all sockets on all boards are supposed to be populated (there's no EPROM #5 on a Defender ROM board, for example). Always question empty sockets.
Next, check for broken devices. Anything that has a higher elevation than the other parts is likely to have been crushed if stacked upon. Trim pots and clock crystals are frequently broken devices (if your board has no clock crystal on it, you know something is wrong :-)). Large capacitors also should be checked. Wiggle them a little bit; they should be firm. If one (or more) capacitor legs are loose and sliding in and out of the cylinder, you should probably replace it. You'll have to look closely to find problems with the smaller devices like broken diodes and broken or burnt resistors. Check for burnt or cracked ICs and transistors, too. If you have to replace an IC, remove the IC and put a socket in it's place first.
Clean off the edge connector (if there is one) using the same method above for cleaning IC legs.
It's ready to be tested!
Tools for soldering
Find a ground and +5VDC rail and ohm them together. They should not be a direct short (0 ohms). If they are, something is very wrong, but finding out exactly what is wrong is the hard part. Do this with the other power rails on the board. I usually hook up my volt-meter to ground and +5VDC points on the board so when I switch on the power, I can monitor the power coming up (and switch it back off quickly if I'm not getting what I expect).