Sorry to hear about your problem. I would start by explaining the difference between one hand swelling and both hands swelling. If swelling is present in both hands it means that there is something wrong with any major system of the body which is causing edema in both hands. If swelling is present in one hand , then the problem is not likely to be systemic or related to some major system of body, it seems more likely to be related to the arm that is involved.
You have swelling in right hand with history of playing computer games for long periods of time. The causes of swelling of right hand can be
carpel tunnel syndrome(in some cases)
Tumor in upper lung
The most probable cause in your case seems to be Tendinitis or tenosynovitis. It is because when we work on computers for long, our wrist is flexed in a specific position for long and the tendons get stretched. However it is not possible to confirm for me the exact diagnosis without examining. I would suggest you to visit a surgeon or an orthopedic surgeon so that he may be able to examine you.
If the diagnosis is confirmed, then you would be prescribed anti inflammatory drugs, to reduce inflammation most likely. I hope I had been a help to you.
If you have any further questions, feel free to ask. I will be more than happy to help you out.
Best of luck...Take care.