Image stabilization might work if the screen is not moving that much – it would do most of the tracking for you by nature of the tool. Right click on the clip and then choose “Stabilize Media…” then follow the prompts. You may want to try a few settings to see if you can get it to work.
If that does not work, then your only other option is track motion or pan/crop keyframes (as rs suggested), or even adjust frame by frame if the movement is serious enough.
You may want to find a way to mount the camera to the gameboy to keep the camera in a fixed position for best results. Or look online to see if there are any gameboy shooting tips out there!
