Hi Cliff,
Most commonly, the issue is a series of bad capacitors. If you can solder and want to get your system back up and running, you'll need to replace all of them. It's best to replace them all as they're low quality capacitors and will go bad eventually, if they haven't already.
You can buy replacement cap kits off eBay that will give you all the ones you need. Take note here: you'll want to know if your Game Gear is a model 'K' or not. You'll see this one the back of the Game Gear on the model. Usually, they're just model 2110. There are a few that are 2110 K.
If your model is a 'K,' then you'll want to buy a cap kit for a VA4 type.
Model: 2110 - Buy a cap kit for VA1
Model: 2110 K - Buy a cap kit for VA4
Once you have that sorted out, you'll also want to buy a 4.5mm Female Torx Security Bit driver to open it up.
Finally, after that's all out of the way, you have your security bit and cap kit. Open the device, lay out your parts, making sure to match the Ohms of the replacement cap to the cap on the board (you should get instructions) and begin swapping out the parts.
This is a guarenteed way to fix the Game Gear and get it running again. I've fixed a dozen of them so far, replacing all of the caps gets every initial issue out of the way.
Good luck, be patience and have fun!