I play a lot of video games and I have a pain in the knuckle of my thumb. It's like a throbbing feeling. What could this be?

GAMER's thumb. A new problem seen in video game players (gamers) is an overuse injury=thumb tendonitis. The treatment is rest, ice, splinting and anti-inflammatories (e.g. Aleve (naproxen) 2 in am, 2 bedtime) hope you are better soon!
Could be ARTHRITIS. Excessive use of a joint may cause "wear and tear" arthritis commonly referred to as degenerative joint disease or osteoarthritis. The erosion of cartilage plus inflammation cause the pain eventually resulting in raw bone on bone as well as bones spurs (osteophytes) which can result in tendon/ligament disruption. Once the cartilage is gone, it is gone and cannot be reproduced with current techniqu.
Video game violence. We wrote an article about this many years ago. It is a type of repetitive trauma. We called it video game violence. This is commonly tendonitis or bursitis. To help this use the heat, exercise, ice treatment program similar to how to help control sports injuries. Heat before the activity and after the activity ice the hand. Sometimes occupational therapy or injection may be necessary.