I suck at coming up with names. I have a fox character I created back in January that is still nameless. What I've been doing is looking up names on baby name websites to see if anything jumps out at me. I've also looked up names of fox breeds and have sorta gotten some name ideas, I just haven't decided on one yet
I either make up a name, search for a name that might work, sometimes (Though not usually) I might ask for an opinion from my sisters or friend, OR what I really like doing is taking names and mixing them up. Like I have a character named Ayllinae, in which I took two names from a random name generator (or was it three? lol, don't even remember) and poof~ Out came the name Ayllinae.
I make sure it's special or has a special significance. It depends on the circumstances, if it matters also to someone else then I take their concerns in consideration too. But if not I juts name away ether made up or mixed.
Oh man... I always have so much problem with thinking names I'm so bad at it, that its scary So I usually just check Old Norse names online and look for one that suits the best, by the meaning of the name and if name sounds good itself.
Sometimes names come to me at random times, sometimes I do research on the name. Sometimes people give me their opinion on the names but mostly my characters all remain nameless until I find something that sorta fits them. I kinda like my last process cause then I can probably get the personality down and the name normally would just follow suite... or it'd be totally random and based of something in real life. xD
When I name a character I either base it off of what they look like or who they are. For example, my character Toaster... He is a robot cat that was designed to make toast. I made him late one night and the name kinda stuck with him. Then I have Mr. Derpinson he is actually based off of the snow golems in minecraft and the name kinda came to me when I was playing minecraft with friends. He has been Mr. Derpinson ever since. I also sometimes look up names that mean certain things.