Zenyatta's colt has not been named. In keeping with the Mosses' practice, he will probably be named while he is in training at the Mayberry farm in Florida (or wherever he is started), just before he goes to the barn of his trainer. (Presumably this will be Zenyatta's trainer John Shirreffs, although the Mosses also use John Sadler.) The colt's owners like to give each horse a chance to develop his/her personality before s/he is named. They take naming very seriously, and have said that choosing a name for Zenyatta's firstborn "keeps us up" at night.
Right now, the colt is known as "12 Zenyatta" or 12Z for short, because he is Zenyatta's 2012 foal. If he has a barn name, it has not been revealed. The Tapit foal that she is carrying will be known as 13 Zenyatta, or 13Z for short.
12Z was weaned last summer and has been turned out in a herd with other weanlings. He and his buddies are due to be moved to another division of Lane's End in December. There he and his friends will be separated by sex and will join their yearling herds.
You can find information about 12Z, along with lots of pictures, at zenyatta.com.
zenyatta.com; David Ingordo BloodHorse interview