TL;DR: Thank ancient Greek and Latin for the confusion, but I pronounce it "ee-ther" because that is actually the correct way.
Although it has almost entirely fallen out of use in the last century, you might have seen the pairs of letters Ae/ae and Oe/oe written as Æ/æ or Œ/œ, respectively (these are called ligatures). This was to indicate a connection to an ancient Greek or Latin word from which the Enlish word was derived. In some cases, the ligatures are simply split into the two separate letters like in the word phœnix/phoenix. In others, the spelling is changed altogether and the word looks closer to the way it is actually pronounced, as in the word dæmon/daemon, which is almost always written nowadays as demon. However, there are words that are written with ae or oe cannot use the ligatures because they do not have Greek or Latin origins (like maelstrom, which is borrowed from Dutch).
Writing aside, the pronunciation varies when you see these letters depending on how it was pronounced in the original Greek and Latin. In the case of æther/aether, the correct pronunciation is with the "e" sound (like "ee-ther"). In these cases where the word does come from Greek or Latin and can be written with the ligatures, ae/oe will never sound like "air," "ay," or "eye" (so "ay-ther" is incorrect).