It depends on the story and the character.
I like to realllllyyyy try and flesh out my character's personality, and sometimes that just helps to figure out a good name. Other times, I could try to do that thing that J.K. Rowling did with Tom Riddle and "I AM LORD VOLDEMORT" but it can sometimes be too predictable or seem way too much like something is odd about the name, and I don't like to have people uncovering things too early on just because of a name.
Usually names for my characters just come to me. If I ever struggle with it though I usually go on those baby name sites and scroll through names until I find one that I feel is fitting. Choosing last names though is a nightmare for me >~<
I'm a bit of a knee-jerk namer. I have a list of names I like but even the most random words become names if they seem like the right fit, doesn't matter how bizarre they are. If two characters are from the same culture I aim to find common ground but even then no rules set in stone. Don't worry too much about satisfying your readers, choose names that matter to you. They're your people. Have fun and get invested.
A lot of my characters names are pretty nonsense, Krisle, Trisc, Lairikk. I just go through the alphabet, and make noises until something sticks. I also tend to look through name lists online (of many languages). There's also a lot of name generators online. I find that they help me brainstorm. So if you see a name kind of like what you want, you can just tweak it, and BOOM you have an unusual name. (I.E. Kris=Krisle, which is basically the name Crystal without a T and spelled weird)
How I named a character recently in real time:
What should her name be?
Sarah? No.
Chris? Nah.
Angela? Ew no.
Debbie? Hmm.
Hmm.
Yeah sure why not.
For me it depends on the story. If it's a new idea I'm just trying to get a feel for then I'll use some placeholder names. When I'm working with an idea that I've been kicking around for a while I'll try to start honing in on specific names for the characters. I like the idea of character names having deeper meanings or connections to common themes in the story but whenever I try to come up with a name like that it comes off as too on the nose.
I love seeing baby names ideas from certain countries, If that helps lol
The others just come to me without me thinking too hard about it 
Depends on the context of the story for me. The first story I made was based on dreams and took place in our universe so a lot of the characters names came from various cultures and all had to do with dreams and psychology. A lot of their names were just baby names I looked up.
As for my current comic I did the same but because it was another universe I kinda just winged it. Like I would google translate words and corrupt them into a name, take preexisting names use them or change them or use places and objects and mix them up. Or sometimes it would just be names I pulled out of my head because I liked them.
I don't have specific names for my characters until I usually draw out how I imagine them. When writing though, the types of people that my characters always interact with are already thought of as I write the story, so all I need to do is incorporate them into the right scenes, even if it means I have to name some as I write them.
I kinda...just go with it? Like while I'm writing, I just put down the first name that comes to my head then change it later. One time, my big, bad villain was named Timmy.
I name my characters after drawing them and after I at least have rough storylines planned.
Regarding how I give names, I really like two-syllable names! Idk why lmao. Also, I have this weird belief that since my drawing style is very manga-like, it's weird for my characters to have English names (lmao). But I also don't want them to have Japanese/Chinese/Korean names since I'm not from those countries. And so, I usually choose names that don't sound completely English/Japanese. Something like: Rina, Noa, Mia, Lia, Karen, Lana, Rena, Mana, Sian (yes, all female names).
I also like English names from different origins:
Tomas (instead of Thomas)
Iren (instead of Irene)
Klara (instead of Clara)
Etc...
If I'm stuck, I usually google baby names XD
Also, if I meet someone/see a name that I like, I tell myself "that's going to be my next character's name" haha.
I name my characters as I write, and I guess that's a bad thing since I write many stories at the same time. People have pointed out I have 2 Wyatts, 3 Christophers, 2 Olivers and 2 Ethans (They're all from different stories.)
Oh well, I'm not one to go for basic names, nothing too extraordinary or out there.
I usually just skim through a baby name site/name gen until I find something that clicks, then I build my characters around that.
It really depends on the story and what kind of mood I want! If Im writing something spooky, I'll name it right away. Fantasy is an iff because I have to search for a name. Or sometimes, my characters have no names because they've been lost through time within their story and is referred by a title.
i give them bad placeholder names until i get better ones and then end up getting attached to the placeholders
alternatively, as with my character jack: forget his name was supposed to be jake and get attached to the Bad Memory Replacement
Much of the time I'll do research. In recent years Behind the Name has been my go to site.
Other times I'll just use whatever name pops up in my head.
But they always have to fit the character that I've already drawn out/envisioned in my head. I don't think I've ever had a name first and created a character around that.
I'm with @Ringarune here - same with me =P
I first create the character, what's he/shes personality? Looks? Life/Past? Then I give them names. I useally do go to http://www.behindthename.com/1
from there I choose the names orgins, how it match with the character. Whats the names orgin?
If I dont find any I like then I combine two diffrent names and make it sound right. That fits more with Fantasy-names
.