Who is Nick? Nobody. There's no Nick hiding inside "nickname" — the word just sounds like there should be one.
An Ekename, Misheard
In Middle English, a nickname was called an ekename — literally "an additional name." The "eke" part is the same old word behind "eke out a living": stretching something a little further than it would naturally go.
The problem was how "an ekename" sounded out loud. Said quickly, "an ekename" and "a nekename" are almost impossible to tell apart — and over generations of speakers, that "n" gradually drifted from the end of "an" onto the start of the next word. The article "an" lost its "n," and the noun gained one: an ekename slowly became a nekename, which eventually settled into nickname.
Linguists have a few names for this kind of shift: rebracketing, misdivision, or metanalysis. Whatever you call it, it's the same idea — a word boundary quietly moves because of how the phrase sounds out loud, not because of anything in the spelling.
Not a One-Off Trick
This isn't a rare fluke. English has done the exact same thing before, just in the opposite direction.
If you've read OddLingo's Apron, Napron article, you already know the mirror-image case: "a napron" was misheard as "an apron," so the "n" jumped the other way, from the noun onto the article. Nickname and apron are the same phenomenon, running in opposite directions — one word gained an "n" it never had, the other lost one it did.
Natural Examples
Her nickname has been "Birdie" since she was six.
The team gave their star player a nickname after his first big win.
Everyone in that family seems to have a nickname for everyone else.
Remember It
There's no Nick in nickname, and there never was. It's "an ekename" — "an extra name" — overheard just slightly wrong, centuries ago, until the mistake became the word.