Just as one shouldn’t offer unsolicited opinions on the name of a new baby or just-adopted child who’s had his or her name changed, one also should respect the pronunciation someone has actively chosen to use for a name. It doesn’t have to be your favourite pronunciation, but since you’re not the one with the name or the one who chose it for a child, it’s not up to you to dictate pronunciation.
To use the examples which spring to my mind first, as a passionate Russophile, there are a number of Anglo pronunciations of Slavic names which are like nails on a chalkboard to me. For example, BORE-iss instead of Bah-REECE, EYE-vinn instead of Ee-VAHN, and Ana-a-STAY-zha instead of Ah-nah-STAH-see-yah. I’ll never understand how anyone gets the lattermost pronunciation out of Anastasi(y)a. If I didn’t know how to pronounce Russian properly, I’d probably think it were said Ann-a-stas-ee-a or Ann-a-stas-ya. But definitely not the most common Anglo mispronunciation!
Anyway, if you know someone is using a different pronunciation from the one you use or prefer, it’s just common decency to use that pronunciation. It’s one thing if you genuinely don’t know, but it’s pretty disrespectful if you’ve been corrected or told several times and keep using another pronunciation. You should always use the name or pronunciation someone has chosen and wishes to be known by. By that same token, you also shouldn’t keep using someone’s original name if s/he’s chosen to legally change his or her name, go by the middle name, or adopt a religious name.
Many of my fellow Lucy Stoners are frequently disrespected like this, such as mail constantly addressed to Mr. and Mrs. Husband’s Full Name, or with the husband’s surname. They’ve told people numerous times they never changed their birth surnames, yet they’re still sent mail and referred to by the wrong name. After a certain point, you kind of have to assume it’s being done deliberately. As badly as I wish more women in the U.S. would keep their birth surnames after marriage, I’m not going to keep using someone’s first surname when she’s changed it and doesn’t regret it.
If it’s not the pronunciation you’re used to, or it seems a little nonintuitive or difficult, all you have to do is ask for a mini lesson. It’s just like asking how to pronounce an Irish name like Aoife, Caoimhe, or Saoirse. The bearer is probably used to such questions and mispronunciations. And really, it’s not that difficult to stop using a pronunciation like EYE-vinn and start saying Ee-VAHN, just as you shouldn’t have to be told over and over again that a J is pronounced like a Y in many European languages, or that a German-born Regina pronounces her name with a hard G.
Asking for the pronunciation is no different than clarifying if someone, for example, spells Philip with one or two Ls, which spelling of Allison or Megan someone uses, or if it’s Elizabeth with an S or a Z. And while I personally dislike certain styles of Cyrillic transliteration, I’m not going to constantly spell someone’s name my own way if s/he’s moved to the West and has chosen to legally adopt a certain spelling. Addressing someone as Aleksandr instead of Alexander, or Tatyana instead of Tatiana, is just as disrespectful as deliberately using the wrong surname or pronunciation. It’s about the other person, not you.
However, I personally would advise using a name in a middle slot if the pronunciation is very nonintuitive to the average person outside of the language of origin. For example, the proper French pronunciation of a name like Genevieve or Camille can strike many English-speakers as pretentious or challenging to remember. It’s a lot different than the Anglo vs. Russian pronunciations of names like Boris or Anastasiya.