"I like your nickname. It works for you."
"I like your nickname. It goes with you."
What's the difference between the two? What sounds more natural for native English speakers?
Neither of them 'works for me', but they do represent the different kinds of semantic changes that English words often undergo.
Starting with 'work for': The verb 'work' has a huge number of meanings. One of these is an intransitive use: "It works! It works!" This means that some action or method has had a successful result.
Phrases like 'It works for me' come from this sense, but have an extra meaning of 'It produces the result I desire'. This is often used to suggest cures for illnesses or problems such as hiccupping: "Stand on your head and drink a glass of water, that always works for me."
In the sentence you give, however, as user162097 says, it now seems to mean something like 'it suits you'; i.e. it is a good match for your appearance or your personality or whatever. This is a new use of 'work for'. It's sort of a semantic extension to 'the result that I want'
In fact, it seems that nowadays just 'work' can have this meaning. In the movie 'Out of Sight' for example, there is a scene where a character who is robbing someone's house picks up the victim's expensive clothes and starts holding them up in the mirror to see if they 'suit him', muttering 'oh yeah, that's working, that's working." To me, this is just weird; maybe I'm old, or maybe I've been out of the States too long.
The second example is not really a new usage; it's just extending the range of the original meaning. X 'goes with' Y means that x and y are a good match, i.e. they are compatible. This is often used with fashions or colors: "Are you crazy? Red pants don't go with yellow socks." I think that 'The nickname goes with you' is using this meaning, as if a nickname were something you might wear. You could say this is a metaphorical use, but I would never say this either. Red pants and yellow socks, yuck.
The first is fairly idiomatic English (though may not be entirely appropriate in this context), the second isn't really. You could say that an item of clothing "goes with you", but I'd be more inclined just to say "it suits you", which you can say about the nickname too. You could also say that "the nickname fits you" or "it's appropriate/apt" or "it sums you up".
"It goes with you" is not used with the meaning "it suits you." It can have a literal or figurative meaning of accompaniment:
You must be fond of your dog. It goes with you everywhere.
That's the trouble with depression. It goes with you everywhere.
There a somewhat archaic usage of asking someone about the general state of things, captured by the more modern "How's it going?" Consider the story "The History of Joseph Senkowitz" by Hannah Berman published in the The Advocate in 1911:
Joshua was the first to break the silence.
"Well, Joseph, how goes it with you?" he asked, seating himself on the large wooden sofa.
"And how goes it with you, master?" Joseph replied in a timid voice.
"It is well with me, it is well. Everyone knows that. And you? You have not told me how it goes with you."
"I am well, I am well, Joshua Aaronson," was the reply.
"You do not look very well. I should have thought otherwise if you had not said you were well."
The phrase can also mean "this is your predictable manner." From the story How it Goes with Me by idradmin published in The Idaho Review, Boise State University's literary journal:
“Do you notice a tendency in yourself to become caustic?” he said. He didn’t wait for a reply. “Here’s how it goes with you,” he said, “if I tell you someone was rude to me, you become rude. If I say that a bartender didn’t do her job, you don’t behave like a friend—you jump to the conclusion that I would find that intriguing. What does that say about you, that you’re so oppositional that you criticize when I need a little sympathy?