Take these suggestions with a grain of salt, as I don't have anything currently posted on this site. Most of my writing has either been for role-playing games (private GMing, not published stories) or for a sequence of stories that I am hoping to eventually do something with, but unfortunately I'm both a procrastinator and a perfectionist. Still, I have done lots of research regarding naming characters, so maybe that will help.
For fantasy or alternate reality fiction, I usually choose a specific world culture and mine in for words, not just names, then apply English grammatical rules to make things more memorable and pronounceable. As much as possible I'll stick to one language/culture within any insular society, as it helps make everything sound connected. For example, if the chosen kingdom is using modified Russian names, I'll use both modern and historical references, but I won't through in a Spanish name unless the character is clearly a foreigner, and even then I'd be cautious.
Fort science fiction and modern stories, I keep in mind three ideas: multiculturalism, laziness, and cultural drag.
Multiculturalism is obvious, and growing in visibility, but still not as well accepted as many progressives would like. Just look at how many people make fun of "silly celebrity names", even when many of those names are just out of fashion or even culturally traditional. Years ago I saw some TV presenter suggest that Kirsten Dunst was trying to sound special by pronouncing her name KEER-stan, rather than the more common KUR-stan; in actuality KD uses the German pronunciation because that's her heritage, and how her parents pronounced it.
Still, it is not unlikely that in the future the acceptance and visibility of a wide variety of names will only increase, so that in a few hundred years the idea of culturally inappropriate names will only apply to a few nations with closed borders and in history books.
Laziness refers to the fact that, over time, people will try to make things easier for themselves, and that includes language and names. In Old English (pre 1066 AD), there were a lot more long vowels and hard consonants than we use today. That's where we get a lot of silent letters from (eg, in "knight", not only would the "k" be pronounced, but the "g" and "h" would be also, similar to the German "nacht"). Middle English (11thC - 15thC) had a lot more in common with Modern English, but was still a separate language.
For naming, laziness means that people will simplify names both over time and over usage. How many people do you know who insist on using Christopher, Daniel, or William in day to day life, versus Chris, Dan, Will, or Bill? How many Catherines go by Cathy, how many Elisabeths are know as Liz?
Cultural drag refers to the fact that some cultures resist change more than others do, and this affects language as much as it does economy, technology, and politics. For example, the Russian and Polish languages separated about 1,500 years ago, but I have been told that if a Russian speaker and a Polish speaker -- each knowing nothing of the other language -- speak to each other slowly they can understand each other.
The Russian name Nadyezhda (roughly NAD-jyez-da, but I suck at Russian) is the long form of Nadia. In Russian culture, especially historically, it would be a grave insult to use the short form of a name without being intimate with the person. For a woman, that would mean her parents, favoured siblings, and her husband.
China likewise has a lot of cultural drag, though for very different reasons and in different ways. In some villages, all children of the same generation are given the same middle name. In Vietnam there are about a hundred family names in total, so it is not uncommon for two unrelated people of the same name to marry.
There was actually point of that huge wall of text, other than the fact that I only say something if I really have something to say. If you're just writing a short story, set in some vague location near the present, then just choose names randomly, using any of the suggestions or resources above.
If you want to write a story taking place anywhere else in the world, or more than about 25 years into the past or future, or if you're planning to write a long story or series, then you may wish to actually think about how the pieces all fit together before you start naming characters.
Personally, I often name characters as the last step in writing. In fact, I current have a short novel that I'm working on in which I have created two cultures, have the plot laid out, have written over 50 pages including moving the action through three locations, and I still haven't named the main characters or even the first-person point-of-view protagonist. The characters' personalities are all set, but I use place-holders for the names, because the names are important to me.
Obviously I don't expect anyone else to go that route, we all have our own styles, and our own quirks, but my point still stands. That point, in case it got lost in the ramble, is that you need to decide for yourself how important names are to you, and if you want them to be consistent both within and across stories. If you are planning multiple stories, or hope to draw a large audience, some extra forethought before writing will make a big difference later.