There is no "nick" in nickname. There is a mistake, repeated until it became the word.
The word began as ekename around 1300. The first half is eke, from Old English eaca, "an increase," so an ekename is an "added" or "also" name. The same eke survives in "to eke out," which first appears in the 1590s.
In Middle English you said "an ekename," with the article an before a vowel. Speakers heard the n at the join and moved it to the next word, so "an ekename" became "a nekename." By the middle of the 1400s the form neke name is recorded. The spelling later settled as nickname.
Linguists call this a misdivision of a word boundary, and English has several others. Merriam-Webster lists several, some going the opposite way, with the n lost to the article. Apron began as napron, from Middle French naperon, a cloth cover. The snake adder began as nadder, from Old English nǣdre, and the n-less form has won since about 1500. An umpire was once a noumpere, from French nompere, "one without equal." A newt gained its n the way nickname did: it was once an ewt, related to eft.
The meaning shifted too. A nickname is now a familiar or pet name, but Etymonline notes that the word often described a name given in mockery or reproach. The verb "to nickname" appears in the 1530s.
So when you call a friend "Mike" or "Bubba," you are using a two-step slip: a word meaning "extra name" that picked up an n from the article before it.