All I can say is that I have only ever heard the word with one pronunciation - but I have travelled through Noth America, the UK, all through Europe, and Australia and New Zealand.

Whenever anyone has used the word (which is admittedly not often!), it has always been pronounced "ANT -o- nim" - the insect; the surprise; and... and... Have you read Robert Jordan's The Wheel of Time? The "Nym".

EDIT:

Actually, there is a word "autonym".

Roughly speaking, the suffix "-nym" means "word". Hence:

• "anti-" means "opposite", so an "antonym" is an "opposite word":

  "cold" versus "hot";

• "syno-" means "same", so a "synonym"" is a "same word":

  "hot", "scorching", "boiling";

• "homo-" means "like", so a "homonym" is like another word, but different:
  • A "homophone" is a homonym that has the same sound:

    "sea" versus "see";
    "write" versus "right";
    "there", "their" and "they're";
    "pair" of slippers versus "pear" of fruit;

  • A "homograph" is a homonym that has the same spelling, but maybe a different pronunciation:

    "tear" in the eye versus "tear" in clothing;

  • And of course English is full of words that have both the same spelling and pronunciation - yet mean different things:

    "lie" (down) versus "lie" (untruth);

Which leads me to "autonym":

• "auto-" means "self", so an autonym is either:
  • A word which describes itself

    "Autonym" describes itself, so it is an autonym;

  • Referring to yourself by your country or language:

    "I am Dutch";
    "I am an English speaker".

But somehow I don't think the person saying "AUT -o- nim" was talking about self-words...

Cite: http://www.magickeys.com/books/riddles/words.html