Strahd's name is of generic Eastern European inspiration. He rules Barovia, his brother is Sergei, the woman in their love triangle is Tatyana — all extremely Eastern European names by the standards of Midwest America in the late 80s.

So, there are some points of commonality among Eastern European languages — or, more to the point, points of commonality in the American stereotypes of how Eastern European names are pronounced and written. (They're not actually that inaccurate, based on observed naming patterns and all, but the name almost definitely came to the game via “what sounds suitably Eastern European and vaguely vampiric?”, not rigorous linguistics research. So, 80s American stereotypes about Eastern European names is it.)

First the easy part: von is a nobiliary particle that attaches a noble's given name to their House name. (Like Madonna, nobles only have one name!) That makes Strahd specifically “Strahd of House Zarovich”, not to be confused with some other Strahd. It's spelled a bunch of different ways depending on the language, but this particular spelling is from German. So, it's easy enough to pronounce von — you can go Movie German Accent and just pronounce it how it looks (which is probably how the designers pronounced it and how your players will expect it to be pronounced), or if you want to get slightly more technically accurate, make the v really soft, almost an f sound, like “fon”.

Zarovich should be pronounced with an English ch sound. -vich/-ovich is a common suffix on surnames in Slavic languages, which means “son of”. It's interchangeable with -vič/-ovič, that háček on the c literally meaning “pronounce this which a ch sound.” (So Strahd's dad should be named Zar? … Not a very Slavic name, that, but I bet they didn't realise that's they were writing a patronymic and just kinda made it up to sound “vampire-y.” Nevermind that modern ideas of vampires came from Hungary and Hungarian is a completely non-Slavic Eastern European name. Oh, and it may be a coincidence, but it's cognate with tsar or “king”, which would make more sense, assuming they put research into the name instead of random Eastern-y syllables. And it's also after a “von” so it's not his dad's name/title, Zar is from his great-great-great-…-grandfather, who founded a dynasty named “Son of Zar” or “Son of the Tsar” and it calcified like that. As names are wont to do. But I digress.)

To pronounce Zarovich right, just give it a bit of a movie Russian accent. Yeah, I know, a bit of a letdown, isn't it? But apart from Slavic names having sometimes a tongue-twisting number of syllables for a typical English speaker, the pronunciation is often pretty straightforward.

Strahd is the hardest part to nail down an origin for with certainty, because it seems to be completely made up just to sound and look good on a vampire lord. Making up names was really uncommon in Slavic languages until recently (there are very few first names in Slavic languages and they just get recycled a lot, because the Orthodox Church didn't like pagan names and for centuries mandated everyone taking the names of Saints), and it looks really Germanic, so we can go with the trend set by the “von” and say it is a German name, albeit made up.

In German, Strahd would be pronounced with a very long, “aaaah” sound, like an upper-class British accent would say it. The str- is a bit fussy though, when said by/to English speakers. Depending on whether the designers were being accurate when figuring out his name's canonical pronunciation, it could have an authentic German pronunciation (“shtr”), or it could have a lazy American approximation (“str”, the way it looks). To English speakers, the accurate pronunciation will be more foreign and harder to say without practice — the foreignness might be a pro, or it might suffer the fate of funny/hard-to-say names in RPGs and get mocked by the players. Since presumably your players will be using his given name more than his full name, you may want to simplify it to the English approximation of a plain “s” to avoid that fate and maintain the Count's gravitas. (Even if you say “Shtraaahd”, they might say it “Straahd” anyway, too.) It's your call, as you know your players best.

So, most accurately of the points considered, Shtraaahd fon ZAAHrovich.† I vant to suck your bluuud. Etcetera, vampire noises, etcetera, with Eastern European aristocratic gothic aesthetics.

It's not 100% accurate, but we've put far more thought into the name by now than I suspect the designers ever did, and it's far more authentic-sounding than Straahd von ZAAHrovich—which for most audiences would still be just fine.

^{† Before the students of linguistics in the audience object that this should be in IPA — if you already know enough about languages to be able to read IPA, you don't need to ask how to pronounce Strahd's name!}

Officially THAK-oh or THAKE-oh is acceptable

This D&D FAQ on the Wizards of the Coast Web site has a section entitled How Do You Pronounce...? that says

Here are some commonly mispronounced words and their dictionary pronunciations where they are available and common-practice pronunciations or TSR rulings where they not. For more general pronunciation help, see the article "Ay pronunseeAYshun gyd" by Frank Mentzer in Dragon #93 (Jan. 1985).

It has the following entry:

THAC0: either THAK-oh, or THAKE-oh

So you and your friends were both right.