A friend later on bought me an Arabic "wild thyme tea", which is labeled as زعتر بري in Arabic. According to Wikipedia, this is thymus serpyllum, which is different from the culinary thyme that we use in the west (although Wikipedia says that kind of thyme is used for tea in Armenia). One more potential source of confusion is that Wikipedia's article on za'atar says that زعتر بري could also be another Middle Eastern thyme (thymus capitatus), or even origanum vulgare (oregano)!
So I'm not positive exactly which herb is in this Arabic tea, but I do like it better than the thyme tea that I made for that party.
Edit: it's this one: http://karabetian.com/food/tea-and-coffee/tazah-wild-thyme-t..., although I'm not sure I would be that comfortable giving my credit card to this web site. I'll see if it's available in my local Arabic market.