Nothing is as irritating as a noisy toilet. Fortunately, a toilet has only a few moving parts. And usually, there is only one of them that will make noise as the toilet tank nears the full mark. That's the ball-cock valve (water inlet valve). Interestingly, this valve is operated by the "wire" and "ball" that you refer to in your question. The wire is actually called the float arm and the ball is known as the float ball. During the flush cycle water travels from the tank into the bowl. As the water level in the tank drops, so do the float ball and arm. The dropping float arm opens the ball cock valve, letting water into the tank to refill it. As the water level nears the top of the tank the reverse happens _ the float and float arm begin to close the ball cock. This is where the noise comes in. A dry or deteriorated ball cock valve gasket can cause a vibration in the valve. When this happens the arm and ball might vibrate as well. Your repair is simple. Either replace the gaskets in the ball-cock valve or the valve itself. Replacing the gasket is less expensive, but replacement of the entire assembly is an easier task and is probably the least expensive proposition in the long run.