So, I have found that about half of the game gear games I purchase have trouble booting. I can always tell before I power on the system because the games that won't boot fit in way too loose. The ones that always work still fit snuggly.
A few months ago I saw a collector show his method of fixing game boy games that don't want to boot. He then proceeded to bend the game a bit. It sounds weird, but the odd thing is it kind off works. Basically, his explanation is that the contacts and board on the games compress over time and use and then no longer get good contact with the pins.
I recently decided to put a tiny fold of paper between the circuit and the game case (touching the side of the circuit that is blank). This pushes the circuit a tiny bit closer to the pins.
Now every game boots first try.