Here's a little troubleshooting FAQ I whipped up.
Batteries won't power it on?
So you just got your fancy DMG on the cheap! You get excited, pop in some batteries, grab a game, and... nothing.
First we have to make sure it's not just loose batteries. Try rotating the batteries, and see if it flickers to life. If it does, that means the battery contacts are probably oxidised or dirty in some way.
To fix this problem, simply take rubbing alcohol and scrub it on the battery contacts. If this doesn't fix the problem, take it apart, and reflow the solder joints
If the batteries don't turn it on, then try using a power adapter. If that doesn't work, there's probably a fuse to bridge or replace, and once this happens to me I'll write it in the guide but I've never had to 
Games won't load?
So it powers on, but all you see is garbled text or a black box scrolling down the screen. This is pretty simple - the cartridge connector is dirty. Take a Q-tip, and remove most of the fluff (but not all) and dip it in rubbing alcohol. Basically shove it down there, let it spread out a bit, then take a cartridge and insert and take it out a lot of times to help rub it in. Wait a few minutes for the alcohol to dry, and try again.
No sound at all?
If you hear no sound, it could be a few things.
1) The speaker is physically broken
2) The speaker is disconnected
3) The game boy thinks you have headphones plugged in, and has shut off the speaker.
4) The volume dial is damaged or dirty
Let's try to rule out #1 and #2 first. When you turn on your game boy, listen for a tiny pop sound out of the speaker. If you hear it, #3 is the problem. For this, take a Q-tip, remove some of the fluff, douse it in rubbing alcohol and just shove it in the headphone jack, wiggling it around. If this doesn't fix it, you might have to replace the headphone jack if you want the speaker back.
Now, for #1 and #2, make sure the DMG produces sound at all.
Put on headphones, and see if you hear anything. If you do, then take apart your DMG. Remove the 6 tri-wing screws (or, in some earlier models, phillips - I've had six like this so far), and take the front half off by firmly pulling the ribbon cable out of the slot. Now, look at the speaker and the two yellow wires. Make sure they aren't severed, and are firmly attached.
If they aren't connected, simply resolder them. If they are, then it's time to test out another speaker, as this one might have problems.
Now, for #4. This one is a bit more annoying, so it gets its own step.
Volume isn't consistent, a left/right channel is missing, or no sound comes out at all?
This is probably an issue with the volume potentiometer being dirty or damaged.
You can get lucky by dabbing some rubbing alcohol on the pot and rotating it to spread it around, and this often fixes it. If that doesn't fix it, you might need to replace the pot. Underneath it is the resistance value you need; it should be easy to find a compatible one.
Buttons aren't very responsive or hard to press?
If a button doesn't work very well or at all, you probably should clean off the conductive pads that make the buttons work. Take apart your DMG as mentioned above, and from the front half remove all the phillips screws so you have access to the front board. Take rubbing alcohol (surprise surprise) and swab off the conductive spots where the buttons go, and also the rubber button pads with the hard black conductor underneath. Reassemble if it's all good from there.
Screen has vertical lines on it?
If your screen has vertical lines of missing pixels, this should be an easy fix. Take apart your DMG to the front board as described above. The screen, the large, green-yellow glass square, has a rubber strip below the viewing area. Remove this, and the small double-sided adhesive strip below it. Be careful not to dislodge the thick brown cable underneath it. Take a soldering iron, and firmly press it (while hot) on the ribbon cable below and around where the bottom of the line is. Over a few minutes of this, it should go away. Others might flicker in and out during this; don't worry about that or if the screen gets discolored, as it should return to normal a bit later.
Screen has horizontal lines?
This is a placeholder, I've had very limited success. More info later on.
I'll add more as people describe problems!
