Please excuse the Latin mini-lesson that follows. Besides being a thayeri addict, I'm a science educator and I took several years of Latin in high school and in college, so I can't help myself.
By today's conventions, it is properly pronounced as Thayer-eye. In classical Latin (not church, or ecclesiastical Latin) pronunciations were quite different, The letter "i" would have an "ee" sound as in timor (TEE-more), Latin for fear. Also, the letter "v" was pronounced like a "w", which means that Julius Caesar's famous phrase "Veni, vidi, vici" (I came, I saw, I conquered) sounded like this: WAY-nee, WEE-dee, WEE-kee. Sounds Elmer-Fuddish to the modern ear, but there you have it. As Christianity spread throughout Europe and a central church emerged, Latin was used as a common language that people from all around the former Roman Empire could understand. Naturally, both pronunciation and vocabulary changed over the centuries, and today most people are familiar with "church Latin" in which "v" is pronounced as "v", "c" is pronounced as a "k" or a "ch" (as in church) depending on what vowel follows it. Catholic priests would pronounce Caesar's famous phrase as "VAY-nee, VEE-dee, VEE-chee".
But besides classical and ecclesiastical Latin, there is yet another set of pronunciation rules developed for the Latin naming system first proposed and used by Linnaeus. It is in this system that a terminal "i" is pronounced as "eye".
So then, back to thayeri. The case is, as they say, that "the eyes have it." (Sorry for the groaner.)
