"Breakfast ball" is just a "mulligan" for the your tee shot early in the morning.
What is a mulligan? Well, it is basically a do-over for your first stroke. All of us have done it before, taken your first shot and seen it arcing to the left; annoyed and a little embarassed, you just place another ball and take a correction shot. A breakfast ball is essentially the first shot you take on that day at the golf course and redo it due to your lack of warm-up (of course).
This term further extends out to other meals of the day, the next one being the "lunch ball." Taking a lunch ball is essentially taking a mulligan after you have just come back from lunch. A relaxing lunch with friends before you play a round of golf is sure to put you in good spirits. Who would be opposed to a lunch ball then?
Breakfast ball in official matches
There is no provision in the official rules where a player can hit two off the first. Breakfast ball is bound to casual, informal matches between friends on the golf course. It is also easier to just do a mulligan that to go around searching for lost balls.
One should always ask their partners if it is okay to take a mulligan or any of the various "meal balls." A do-over should not hinder the flow of the round and certainly not be done when people are playing seriously. A good alternative to mulligans are using a handicap score.