on today - is on necessary?

I'm a middle school teacher and my principal always says "on today" or "on tomorrow". For instance, "We will have a test on tomorrow". Is this correct. "On" seems to serve no purpose. Also, is it correct to say, "I will see you on Friday or I will see you Friday?" or does it matter?

I found that two of my previous principals (originally from the NC area) also used the "on today/on tomorrow" structure. I wonder if it is a regionalism?

Hmmmm... That is interesting. To my ear, the "on" sounds natural when it's used with a date or with a day of the week, but it sounds unnatural when it is used with "yesterday", "today", or "tomorrow":
"Yesterday, December 7, 1941—a date which will live in infamy—the United States of America was suddenly and deliberately attacked by naval and air forces of the Empire of Japan." ~ Franklin D. Roosevelt
"Whereas, on September 11, 2001, acts of treacherous violence were committed against the United States and its citizens..." ~ Public Law 107-40, Authorization for Use of Military Force
"The terrorist attacks on September the 11th were a turning point for our nation." ~ George W. Bush
I'm sure that there are plenty of counterexamples, though.
"On today" (and its close parallels "on tomorrow" and "on yesterday") sound wrong. With a day of the week or other short description such as "Thanksgiving" it's possible to omit the "on", but some people get all het up when they read a sentence like Leading economists announced Tuesday that the recession is nearly over. They claim it's a "garden path" sentence; that it makes them think that the economists announced a day of the week.

This is a pet peeve of mine. We have a local weatherman who consistently says "on yesterday" and "on tomorrow". I had never heard this before and everytime he says it I bristle.
He also has a habit, when speaking slowly, of saying "ah" before certain words. I have heard another reporter on local station do the same thing. It's not a hesitation, not like they don't know what to say next, but actually sounds like they were trained at the same school of enunciation! The "ah" often feels like a precursor to the next word, an emphasis of sorts. It doesn't really help. In fact, I always think he is going to say a word that begins with the letter "a", then he says the word "school" or something! "The children were excited to be leaving aaah-school today".