The correct response to "top of the morning to you" would be "and the rest of the day to you." This colloquial Irish saying is used in the morning to greet another person.
The meaning of this greeting is to wish that someone makes the most out of the morning. A typical response would wish that person a good rest of the day. Even though the phrase is not commonly spoken nowadays, it is instantly recognized as something that is uniquely Irish and characterized as such. The greeting is used in many marketing and advertising campaigns to promote Irish culture for tourism purposes.