You should know that when you stare at the shining screen with few blinks, your eyes get very dry and will feel very uncomfortable, thus, your biological mechanism gives your eyes order to produce some water to keep them moist. So, what you should do is to take breaks after playing games for 30 minutes, better yet, get yourself some eye drops to relieve your strained eyes.
It is very likely that your eyes become fatigued which a common cause of watery eyes is. It means that your eyes need a break. Looking out a window at something distant regularly is the best way in my own personal experience. Watery eyes can also result from a symptom from some type of allergies, a sinus infection, headaches, irritations, or even problems with eye-wear such as contacts. In your daily time, you should keep your home clean from any dust, dander, and household plants to prevent your eyes from allergies. Each problem is different, and should be investigated individually. So once it becomes serious, you should go to see a doctor right now.
It must be caused by a long time use of eyes in playing video games. As we know, the eyes water because of the excretion of tear glands in the eyes which can be controlled. If your eyes get high stress with no rest, the inner nerve system in the eyes is in disorder, including the tear glands. Thus, the eyes can't help watering. You should stop playing video games and let the eyes have a good rest.