The expression `ding-dong’ refers to the sound that a bell makes; the back and forth motion of the knocker hitting a bell.
When you say that two participants were involved in a ding-dong battle, what you mean is that they were involved in a long drawn out battle. The fight was intense and there were times when each participant seemed to have the upper hand in turns. In a ding-dong battle, it is often very difficult to predict who will emerge the winner; sometimes, the result is inconclusive.
Another expression which has more or less the same meaning is `see-saw battle’. When you sit on the see-saw, you are up one moment, and down the other.
Example: *The fans were hoping for a ding-dong battle. They were disappointed when the Aussies thrashed the Indians.