Greeks go by the original "arghirotipi'a", foreign languages borrowing the term would go by the phonetic rules of each dialect, so I imagine there isn't a set way to pronounce it.
The inventor of the process, Mike Ware, pronounces it "Ar-gyro-type" (as in gyroscope). The root is taken from Greek, (argiros = silver), but the modern Greek pronunciation is near impossible for English speakers to master (because of a soft Gamma).
It could be argued that the original loan-word argiros is taken from ancient Greek, which had a hard pronunciation of the letter Gamma, making "Ar-gyro-type" a rather correct pronunciation.
But tomato is an American word. Well, Central America. As in Aztec. From Wikipedia, "who called it xitomatl (pronounced shi-to-ma-tlh), meaning "plump thing with a navel". The word tomato comes from a word in the Nahuatl language, tomatl."
We're running an exhibition of several alternative processes including argyrotypes in Brighton, UK, until November 14th 2008. For details see: http://www.siderotype.com/fringe.htm