As a teacher to thousands of teenagers over the last 15 years, I have heard "bored" more times than I can possibly recall. "Bored" seems to be a multi-purpose word to describe anything that a kids does not like. I have always told my students that it is better to be bored than excited if "excitement" means what it means in a lot of countries--running from bombs and gunfire, foraging for food to avoid starvation, etc. I also have told them that if they are bored that they should recommend a "fun" activity to me in writing. (I've known kids who've started their own investing companies, so teenagers are old enough to take charge of their lives, e.g. find healthy ways to address their boredom.) If doable, I will replace one of the "boring" activities with one of the suggested "fun" activities. Have you considered allowing him, possibly with friends, to try a very wide variety of extra-curricular activities, such as sports he's never thought of trying, following the stock market--websites and books for teenagers on the subject are widely available, or participating in a volunteer activity (of his choice)? Some of my students and classmates have done very interesting things that I wished I had thought of doing when I was a kid. I would suggest trying a few of these activities before he settles on playing video games a 100 hours each week. Having said that, although I'm not a video game fanatic, I can't say that video games are all bad. One of my classmates in business school has done quite well in his video-game related career, which he got into because he loved playing video games.
In deciding how often to let a child get together with his friends, I would ask myself a few questions. Is my child getting enough sleep? Is he allowing himself enough time to do the best he can academically? Is he helping around the house? Is he staying fit? Is he participating in family (your family's) activities as often as you think appropriate? What are he and his friends doing? Are they doing something healthy or unhealthy (vandalizing property, using drugs, etc.)?
It seems that a lot of kids become less bored as they switch from middle school/junior high school to senior high school, so you will likely hear "I'm bored," less and less over time.