Almost all hiccough cures are based on either overwhelming the vagus nerve, or interfering with breathing.
Overwhelming the vagus nerve with another sensation
The vagus nerve is a cranial nerve that sends sensory information about the stomach and other the body’s organs to the brain. When it has been distracted by overwhelming information of another sort, it basically tells the brain that something more important has come up than hiccuping, so it stops. Vagus nerve stimulation has also been successfully used to control seizures in epileptics.
Interfering with breathing
When breathing is interrupted, this increases the amount of carbon dioxide in the blood, which then causes the body to concentrate on getting rid of the of the CO², rather than making hiccups.