Reddit mentions of E-Z Lok 400-M3 Threaded Insert, Brass, Knife Thread, M3-0.5 Internal Threads, 0.375" Length (Pack of 10)

Well, I have some good and some bad news for you.

The good:

• This is a Majesco Game Gear which has a better screen than the originals and doesn't suffer from the bad capacitors issues!
  
  The bad:
  
  
  
• Some Majesco Game Gears didn't come with the rear screw point and some did. You unfortunately have one that didn't.
  
  Here are some pictures of my two Majesco Game Gear systems, one with the screw hole and one without.
  I opened up the console without the screw point to see the inside and the brass screw insert is not present.
  
  One possibility that you could try is adding the screw insert your self as it is a simple threaded brass insert. You would need to open the Game Gear's case, drill out the plastic in the center of the hole and screw in the threaded insert. The insert linked above is just an example and not the correct size. I don't know the size or thread pitch off hand but it would be easy enough to figure out by taking the attachment screw from the PowerBack to Home Depot / Ace / Lowes or another hardware store where they will have a screw sizing jig. Then just buy an insert of that size at the same store!
  Edit: spelling/formatting