Let me have a shot! zu Mittag - here Mittag is a certain time. zum Mittagessen - you go to a place. In this context zu/zum is rather irrelevant when you speak to people since Mittag essen and Mittagessen is phonetically hard to distinguish.

Ich finde hier keine Unterschiede..

Was ißt du zu Mittag? = What do you usually have for lunch? Was ißt du zum Mittag? = What are you having for lunch today?