It can certainly be mortifying to learn you've been blithely mispronouncing someone's name—especially after an extended period! It's even more awkward if you then discover you can't articulate sounds in their native name, and will just have to conciously continue to mispronounce it in some respect.
So, as you've recognized, it's best to be mindful of the issue and work on it from the start of the relationship. How you can do this smoothly depends on the context and personal histories and cultures of both you and the person whose name you're trying to pronounce.
As a linguist by training, I've always had an interest in this—I like to try to pronounce "difficult" names as correctly as possible, and find the rules different languages have about use of names fascinating.
Still, navigating this situation requires diplomacy and tact as much—if not more—than linguistics. Here's what I'd keep in mind:
Your acquaintance's relationship to native English speakers in general
People with names like you describe will generally fall into one of two groups:
Those who have experience dealing with native English speakers, and are well aware of the difficulty.
These fall into two subgroups:
a. Those who are understanding, patient, and generous about mispronunciation of their names, or:
b. Those who are not — perhaps because
They have come across enough native English speakers who can pronounce their names (maybe from travel to other parts of the anglophone world where familiarity with the sounds is greater) that they think anyone else should be able to as well;
They think that you could pronounce the name but are essentially being more-or-less consciously xenophobic by "refusing" to pronounce it (particularly if you're American);
Their own language doesn't make distinctions in the same way English does, so they don't notice anything wrong about your pronunciation, even though it obviously sounds "wrong" to other native English speakers;
- They're just not cooperative in nature.
- Those who are new to dealing with native English speakers — for instance, this is their first trip to an anglophone country, and they have previously only spoken English with speakers of their own native language.
The priority your acquaintance puts on pronunciation of their name
People with the same native name and language can differ in how much they care, and about what.
- Some would like each person to do their best at pronouncing their native name, and will coach and correct you, expecting you to get better at pronouncing it as you get to know them better.
- Some put a higher premium on everyone in the group using the same pronunciation, since it can feel odd for different members of a group to all call someone by slightly different names.
- Some prioritize smoothing out social awkwardness, and will do whatever's necessary to avoid that awkwardness — whether that's selecting an adopted English nickname, or letting each person pronounce it however they happen to.
Your linguistic abilities
In your question you mention what perhaps seems the most obvious and vexing issue in pronouncing an unfamiliar language's names — sounds that don't exist in English. And indeed, this can be a problem. If you can't trill, you won't be able to pronounce a name with a trill, no matter how much you want to. Unfamiliar vowel sounds, clicks, or glottal and pharyngeal sounds will all be very difficult to get right.
But while obvious, "weird sounds" aren't the only, or even the most common, issue native English speakers have with "difficult" names. Also possible:
- Sound density and flow: in English we can cluster consonants together, we can put vowels next to one another, we can start or end words with either single consonants, consonant clusters, or with vowels. Many languages are far stricter about this, and a secondary effect of this is that words (and names) tend to get longer, because less possibilities are available for each syllable.
- Stress: in English, we prefer certain stress patterns to go with certain combinations of vowels and consonants, and there are some vowel sounds we pronounce only stressed, or only unstressed. Other languages do not follow these same patterns, and we may not even be able to hear the difference unless our attention is drawn to it.
- Syllabification: in English, we break syllables in a certain way, and we usually don't even realize we're doing it. Have you noticed how often apostrophes appear in science fiction names? A bug-eyed villain named "K'Onyak" just sounds alien — and not at all like a certain expensive French brandy! — just by forcing syllabification into an unexpected place.
(Stress and syllabification can really matter. Consider the English: I went into the dark room vs. I went into the darkroom. The only difference is in stress and syllable breaks, but we can definitely hear the difference between a dimly-lit enclosed space and a photographic developing lab!)
The bad thing about all three of these is that you must be conscious of them or you may not even be aware you're badly mispronouncing a name, even after mastering all the "weird sounds".
The good thing about these three, though, is that unlike those trills and pops, you can get these right with just a little practice. For example, many North American English speakers have difficulty with subcontinental Indian names that seem very long and complicated, but it's generally just because it's a lot to take in at once and we're not used to it. Get the person's name in writing, and practice it alone until you figure out the rhythm, and you'll find "Nathavarthi Paramparil" isn't really a "difficult" name at all—you just needed the time to assimilate it.
The other English speakers around you
You may not speak their language, but you may speak another second language that has sounds closer to the ones in their name. If so, you may be able to produce a closer approximation than most native English speakers.
But even if you speak their own langage well, you shouldn't assume they want you to address them using native pronunciation. It can sound pretentious to others, causing social discomfort, and may result in the other English speakers in your group trying to match your performance, butchering their name in new ways every time it's spoken. Your acquaintance may wish you'd stuck with the "wrong" version!
For instance: the Russian name Олег (Oleg) is natively pronounced almost identically to the English name "Alec". If you know this, you may be tempted to call him that. And who knows, that might delight him!
But since Oleg is also an Anglicized name that sounds nothing like "Alec", it could also result in others being confused— Who's 'Alec?' Isn't that Oleg? Have I been getting his name wrong all this time? Or should I pull that person aside and correct them, so they stop calling him Alec? — and Oleg may be aware of this sort of confusion and prefer it not happen, even at the cost of everyone "mispronouncing" his name.
Another possibility to consider: many languages have different pronunciations of names depending on how they're used. Some have strict politeness rules, and in their native language calling them by exactly what's written on their passport—even "correctly" pronounced—would be a faux pas.
Some languages have what's known as a "vocative" form, meaning that their name sounds different when you speak directly to them from when they are referred to in the third person. Imagine something like: Olyegu, would you like some water? Yes? Mary, would you please pass the water to Alec? That could be perfectly pronounced as it would be in their language, but in English it still may come across as odd, since our names don't change their sound based on usage.
How you respond
That background wasn't just a long digression — knowing what variety is possible in names, how people interact with regards to "foreign-sounding" names, and what you're capable of learning to say and what is impossible are the fundamentals you need to know so you can know what it is—and what it isn't—reasonable to ask and to expect.
You don't need to worry too much about case "1.a" above — people with experience, understanding and generosity. They'll help you and appreciate when you express interest in pronouncing their name as correctly as possible. Whether it's by coaching you, giving you an "easy" nickname to use, or giving forebearance to many varied butcherings of their name, they'll help you without too much trouble.
People in case "2" above — people new to being around native English speakers — will probably still be feeling things out and won't necessarily know either what it's reasonable to expect from native English speakers, nor what their priorities are. I'd be very explicit that I want to take as much time as they'd like (but no more!) to practice their name with them.
Also, explaining what the difficulty is—a particular sound, the length of the name, an unusual stress pattern, a consonant cluster that isn't possible in English—rather than just giving them the vague sense that their name is "difficult" — will help in setting expectations. Telling them you'd like to see their name written down, or that you'll need to practice for the next time you meet them, or that there's a particular sound that you know you'll never be able to produce — these are useful to them as much as they are to you.
Perhaps the person will choose a nickname or a slight alteration when they learn some sound is hard for native English tongues. Or they'll have business cards at the ready, even at social events, so people can see the name written down and practice it later. But in the meantime, they can enter their name into your phone, or you can ask if you can take a quick video of them saying their own name so you can practice on your own time.
But the one impulse I've heard that I'd firmly steer you away from: don't suggest a new (nick)name for someone that's easier for you to pronounce. And definitely don't make up a nickname on your own initiative and start calling them by it. That is extremely rude! If they want help finding an acceptable alternative name, they will ask. A person's name is, quite literally, their identity, and you have no right to alter it without permission.
But that brings us to the difficult case, "1.b" above: those who know their name is "hard", but for whatever reason, refuse to help you reach an acceptable middle ground. In this case, you can only try your best, but if by the third or fourth meeting, you still haven't got it and still aren't getting any assistance, you'll just have to shrug and accept that you're going to continue to mispronounce their name, and they're going to continue not to like it—to do otherwise is just going to upset you and/or reward passive-aggressiveness.
Thankfully, in my experience, such people aren't ones you want to spend a lot of time addressing directly anyway!