Hello from JustAnswer.
Pain in the knuckle can be due to arthritis or tendinitis. However, if the pain is significantly greater with flexion, it is more likely to be a tendinitis. Both can occur from overuse or wear and tear. In fact, the overuse tendinitis that most people have heard of is tennis elbow, but you are apparently experiencing it in the thumb.
Ice and ibuprofen are reasonable first steps. Although you do not say how long this has been bothering you, the ice is most important for the first couple days. After that, moist heat can also be helpful, and can also directly relieve inflammation. It also may help to provide support, such as by taping the thumb and wrist. We would usually prefer to tape the thumb in a neutral position, but if you need to continue gripping a racket, the thumb can be taped in a flexed position. An elastic bandage, such as an ACE bandage, can help support the base of the thumb, but does not provide good support for the entire thumb. A splint can provide better support, but would hamper being able to grip a racquet. It also would be good to elevate the hand above the level of the heart while applying ice or moist heat.
If these interventions are not sufficient, then it may be necessary to try a different anti-inflammatory medicine. There are only two available over the counter, with the other being naproxen. It is not that the naproxen is a better anti-inflammatory medicine than ibuprofen, but rather that different medicines may work better for different people. There are many other anti-inflammatory medicines available by prescription, including several that are in different chemical classes (while both ibuprofen and naproxen are in the same chemical class), but that would only be an option if you see your doctor.
If there is persistent inflammation, it also would be an option to use a steroid injection, but this also would require seeing your doctor.
And yes, you are correct that the more that you can rest the hand, the quicker it will improve. If you can transition as much of your instruction to the opposite hand as you can tolerate or by using different methods, such as observation and verbal instruction, it would be better.
If I can provide any additional information, please let me know.