It's an adjectival form of the word "tone." "Tony" is short for "high-toned," which means "of high social rank," or sophisticated, fashionable. Here, it means that the speaker suddenly shifts from his usual, casual speaking style to pronounce the name "Hermés" in a "tony" way. Certainly he is saying "air-MEZZ" and not "HUR-meez" (which is how U.S. speakers pronounce the name of the Greek god). Likely he is not only saying "air-MEZZ" but is trying to imitate French pronunciation. "Oh, look at me, I know about all the latest European fashions and shop at a fancy-schmancy French store." There's a familiar joke in the United States. There is a very popular chain of department stores named "Target," like a bow-and-arrow target. It is a mid-priced store--not terribly expensive or sophisticated, but a step up from Walmart. Very often, as a joke, people will pronounce it "tar-ZHAY," as if it were a French word. The joke is that they are pronouncing it in--well--a "tony" way, when Target is NOT actually a "tony" store. I DON'T personally use the word "tony" myself. From Googling I find that it is from "Whiskey Beach," by Nora Roberts. If the book is set in the present day it may be a more common word than I think.

In this particular context, it suggests that he is pronouncing "Hermes" with its proper French accent-- most English speakers will say it as if it were an English word, so he is attempting to demonstrate his sophistication by pronouncing it in the French fashion. Usually, though, it is as Phil says: an accent meant to convey one's high level of education and standing.

Apparently it means a high toned accent, usually used to describe aristocracy/queens english type accent...... youtube the queen of England speaking and you will know what it means :-). I had to look this up myself as I had no idea.