It improves it drastically, due to people having to look at a screen and press a corresponding button without looking at a controller, and a obvious example that does this is are the Guitar Hero and Rock Band series.
Truly, I believe it helps very little with hand-eye coordination, though it helps some. But if you want an excuse to give to your mom, go ahead and use "Video Games uses hand eye coordination, mom!"...
Video games improve hand eye coordination, and there have been relevant studies that have linked playing video games with being a better driver and also helps you catch more balls than you did befor...
When you play video games to much your eyes get red then the next day in school you cannot see well, therefore you would need glasses if its not survire