Mucous cysts on the fingers can be merely annoying or they can be painful. They are not cancerous or dangerous, so having one doesn’t really do much to your hand or health, except the affected fingernail may become bent or deformed.
If you are not concerned with the cyst and it is not bothering you by causing pain or limiting your range of motion, then discuss with this your doctor to explain why you are refusing surgery. Ultimately, it is your decision.